Short Answer: Life on your planet is doomed.
The closest I could come to your requested 30 minute orbital period without dooming all life on the planet, and while having a reasonable chance of the orbital configuration forming, would be to make your habitable planet be a sister planet or moon with a mass of 1 Earth mass and the other planet with a mass of 5 to 10 Earth, separated by 27,00 to 33,000 kilometers with an orbital period of about 300 minutes.
of course if you are content with your story having a low score on the scale of science fiction hardness you don't have to consider such matters.
If the moon orbits at its Roche limit when life begins on the planet it will not be orbiting at its Roche limit billions of years later when photosynthesis evolves and even later when multi celled organisms live on land.
Tidal interactions between the planet and the moon will change the rotation rates of both objects, and will change the distance at which the moon orbits.
If the moon orbits the prograde direction, the same direction as the planet rotates, and has a longer orbital period than the planet's rotation period, the tidal acceleration will push the moon farther and farther from the planet, as is the case with Earth's moon.
If the moon orbits the planet in a retrograde orbit, opposite to the direction that the planet rotates, tidal deceleration will make the moon gradually spiral inward until it either reaches its Roche limit and breaks apart into a ring or collides with the planet.
Triton, the large moon of Neptune, has a retrograde orbit and it is predicted to become too close to Neptune and be destroyed in a few billion years.
If the moon has a prograde orbit but orbits below the synchronous orbital distance and so has an orbital period less than the rotation period of the planet, tidal deceleration will cause the moon to spiral inwards and eventually reach its Roche limit and break up into a ring or else collide with the planet.
Phobos, the inner moon of Mars, is an example of such a moon and is predicted to be destroyed in "only" a few tens of millions of years.
What is the shortest possible day for an Earth like planet habitable for life?
That is one of the factors discussed by Stephen H. Dole in Habitable Planets for Man (1964).
On pages 58 to 61 Dole discusses the range of rotation periods for a planet habitable for humans. Dole decided that the planet's structure would become unstable if it rotated faster than 2 or 3 hours per rotation.
A moon orbiting with a period of 30 minutes would be far below the geosynchronous orbital distance for even a world with rotation period as low as 2 or 3 hours. So tidal deceleration would make it spiral inward. And you say the moon is at its Roche limit.
I am not certain when in the history of your planet the moon is at its Roche limit, but whenever it reaches that Roche limit the planet will face the breakup of the moon, and since the fragments will still be tidally decelerated they will fall on the planet causing a mass extinction event.
Maybe the moon will be tiny and its fragments won't do much damage.
Nope. You describe the moon as (impossibly) filling the entire sky, and say the eclipses last for half of the orbital period of the moon. It must be very large and thus its fragments will destroy all life on the planet.
But if the moon and the planet become tidally locked to each other the tidal effects will stop, since the moon will always be above one spot on the planet.
To be continued.
Part Two: Make the Planet a Moon and the Moon a Planet?
During the last few decades there has been considerable scientific discussion of the possibility of life on hypothetical large enough exomoons orbiting around giant exoplanets in other star systems.
the majority of stars are dim red dwarfs. A planet orbiting in the habitable zone of a red dwarf star would be orbiting so close that tidal interactions with its star would quickly slow down its rotation until it became d tidally locked to its star. The period of rotation of the planet would become the same as the planet's orbital period around the star.
Thus one side would always face the star in eternal day and heat and the other side would always face away from the star in eternal dark and cold.
That is usually considered to be bad for the chances of life on the planet.
But if a moon large enough to possibly have life orbits a giant planet in the habitable zone the planet's tidal effect on the moon will be greater than the star's and the moon will be tidally locked to the planet instead of the star. Which means that the star will appear to rise and set as the moon orbits the planet and the moon will have alternations of day and night.
As the exomoon orbits the giant exoplanet the side of the moon facing the planet will will also get planetlight during part of the night, light from the star reflected from the planet back onto the moon, and there will probably also be eclipses of the star once a day as the moon lines up with the planet in front of the star.
Thus you will have the eclipses you want.
So having your "planet" actually be an exomoon and your "moon" actually be an exoplanet can in some cases work just as well as having your world with life be a planet orbited by a very large and close moon. And it would be a situation where the two bodies would no longer be moving apart or closer.
Part Three: Around the World in 80 Minutes.
Back in 1872 it was rather new and amazing that it was now possible to travel Around the World in Eighty Days. Thus the popularity of Jules Verne's novel and many adaptations of it like the 1956 movie. So as a kid I wondered when it would be possible to go around the world in 80 hours or even around the world in 80 minutes.
Satellites in low Earth orbit sometimes go around the world in 90 minutes. The International Space station orbits the Earth in about 92.9 minutes.
The Earth has an equatorial radius of 6,378.137 kilometers and any Earth orbit would involve passing over the equator twice in each orbit, so 6,378.137 kilometers is the absolute minimum possible distance from the center of the Earth of any orbiting moon, even ignoring the atmosphere and mountains.
According to this orbital period calculator:
A lunar mass moon orbiting at the surface of the Earth would have an orbital period of 84.07 minutes. Thus it seems impossible for any moon or artificial satellite to orbit the Earth in only 80 minutes.
But if the Earth had the same mass but a higher overall density it would have a smaller radius and thus it would be possible for an object to have an orbital period of 80 minutes or even the 30 minutes asked for in the question.
By messing around with the orbital period calculator, I got a result that an object orbiting an Earth mass object about 3,208.65 kilometers from the center would have an orbital period of about 30 minutes.
That world would probably have to have a radius of less than 3,000 kilometers in order for the moon to orbit outside the atmosphere with an orbital period of 30 minutes.
A world with a radius of 3,000 kilometers would have about 0.4703567 the radius of Earth and thus it would have 0.0104059 the volume of Earth. Thus it would have the mass of Earth within 0.0104059 times the volume of Earth.
So it would have an average density of 1 Earth density divided by 0.0104059, or 5.514 grams per cubic centimeter divided by 0.0104059, or 96.099 times 5.514 grams per cubic centimeter, or 529.89179 grams per cubic centimeter, which is many, many times the maximum possible density of a world which has only 1 Earth mass.
You can mess around trying various combinations of mass and radius (and thus density) of planets with a mass within the range of 0.195 Earth mass to 2.35 Earth mass given by Dole on pages 53 to 58, trying to find a combination with an orbital period as low as 30 minutes.
But I doubt you may find a plausible or possible world with a 30 minute orbit.
To be continued.
Part Four: A Black Hole.
Suppose that there was a world with the mass of the Moon, 0.0123 that of Earth, and the radius of the Moon, 1,737.4 kilometers. And suppose that a primordial black hole, formed during the Big Bang, with a mass of about 0.9877 the mass of Earth happened to encounter that world and eventually came to occupy the enter of it. That would make the total mass of the two combined objects about 1.00 Earth mass within a radius of 1,737.4 kilometers.
Thus a moon could orbit that world at a distance of 3,208.65 kilometers from the center of the two merged worlds with an orbital period of 30 minutes.
The surface gravity and the escape velocity of that moon sized world with a black hole inside would be about 132.05 meters per second per second or 13.465 times that of Earth, and 21.42 kilometers per second or 1.91489times that of Earth, respectively.
That seems like a world with far too much surface gravity for Earthlings to survive more than a few minutes. And the escape velocity might be high enough for it to retain a lot of hydrogen and helium gases, turning it into a sort of a "gas dwarf" planet.
Part Five: A Moon Again.
So reconsider a large exomoon orbiting a giant planet in the habitable zone of their star, tidally locked to the planet. One side of the moon would always face the planet and one side would always away from the planet.
The side that faced away from the planet would experience a normal pattern of light and dark as the star rose and set with the same period as the moon's orbit around the planet.
The side that faced the planet would experience the same cycle of the star rising and setting and light and dark, except that the cycle would be interrupted once a day by an eclipse of the star by the giant planet and once a night there would be a lot of light reflected from the planet onto the moon.
So can the day or month of a tidally locked moon of a giant planet get to be as short as 30 minutes?
Interestingly, the sizes of giant planets don't change very much with increasing mass, past a limit which is only slightly greater than that of Jupiter. Past that mass limit, a planet will become denser and denser under the compressing force of its gravity, and will even begin to have a smaller and smaller radius.
Furthermore, brown dwarfs, objects intermediate in mass between planets and stars, also don't get much larger with increasing mass. They become denser and denser instead.
So planets and brown dwarfs don't get much larger in radius than Jupiter. I read somewhere that planets more massive than Jupiter and brown dwarfs have radii between about 15 percent smaller or larger than Jupiter's radius.
Jupiter has an equatorial radius of 71,492 kilometers, so planets more massive than Jupiter and brown dwarfs would usually have radii in the range of about 57,200 to 85,800 kilometers.
So the question is whether a moon or planet could orbit with a period of 30 minutes and a semi-major axis of more than 57,200 to 85,800 kilometers around a giant planet or a brown dwarf.
At a distance of 100,000 kilometers from a planet with the mass of Jupiter, a object with 1 Earth mass would have an orbital period of 293.8 minutes.
At a distance of 25,000 kilometers from a planet with the mass of Jupiter, a object with 1 Earth mass would have an orbital period of 103.8 minutes.
At a distance of 21,849 kilometers from a planet with the mass of Jupiter, a object with 1 Earth mass would have an orbital period of 30 minutes.
And it would be deep inside the planet.
The dividing line between a planet and a brown dwarf is at about 13 times the mass of Jupiter, or about 4,131.4 times the mass of Earth.
At a distance of 100,000 kilometers from an object with a mass of 4,131.4 Earths an object with the mass of 1 earth will have an orbital period of 81.6 minutes.
At a distance of 50,000 kilometers it will have a period of 28.85 minutes.
At a distance of 51,315.9 kilometers it will have an orbital period of 29.995 minutes.
And at 51,315.9 kilometers it should be inside any planet with Jupiter like or more than Jupiter mass.
The dividing line between a brown dwarf and a very low mass star is at about 75 times the mass of Jupiter or about 23,835 times the mass of earth.
An object with about 1 Earth mass orbiting a maximum mass brown dwarf of 23,835 times the mass of Earth at a distance of 62,100 kilometers would have an orbital period of 30.03 minutes. And maybe a extremely massive brown dwarf might have a radius of less than 62,100 kilometers.
So it is not entirely theoretically impossible to arrange the orbits of two objects so that the star can be eclipsed by one object in the sky of a possibly habitable other object every 30 minutes.
But there are some major habitability problems with this situation which someone should consider before deciding on using it.
Part Six: My Recommendation.
I recommend that you give up the idea of having the moon orbit every 30 minutes.
Instead use a planet which rotates every four hours or 240 minutes, a little slower than the top safe rate of 2 to 3 hours mentioned by Dole. And have the moon orbit at the synchronous orbital distance.
Make the planet about 1.5 times as massive as the Earth, for example, and make the moon a twin planet, about 1 times the mass of Earth.
Presumably the sister planet moon formed formed just above the synchronous orbit when the planet rotated a bit faster. The tidal interactions rabidly slowed the rotation of the planet to `four hours and pushed the orbit of the sister planet out to the four hour synchronous orbit and they became locked together there.
At a distance of 17,400 kilometers the two worlds will have an orbital period of 240.8 minutes, or four hours. That seems like much too close a distance to be the result of the worlds moving apart in the past.
So make each world have 1.75 the mass of Earth. At a distance of 19,400 kilometers they will have orbital period of 239.58 minutes.
If each world has a mass of 1.9 Earth mass, they will have an orbital period of 240.67 minutes at a distance of 20,000 kilometers.
So maybe you should make the synchronous orbital distance 5 hours or 300 minutes.
If each world has a mass of 1.9 Earth mass, they will have an orbital period of 300.7 minutes at a distance of 23,200 kilometers.
Make the habitable world 1 Earth mass and the other world 5 Earth masses. they will have an orbital period of 300.4 minutes at a distance of 27,000 kilometers.
Make the habitable world 1 Earth mass and the other world 10 Earth masses. they will have an orbital period of 299.8 minutes at a distance of 30,000 kilometers.
And I think that having your habitable planet be a sister planet or moon with a mass of 1 Earth mass and the other planet with a mass of 5 to 10 Earth, separated by 27,00 to 33,000 kilometers with an orbital period of about 300 minutes is the closest I can come to a reasonably plausible version of the requested 30 minute orbital period.
After finishing my answer I see that the question has been changed to a habitable moon at its Roche limit orbiting a giant planet with an orbital period, and thus day, of 90 minutes.
And according to the orbital period calculator, the moon would have to orbit the center of Jupiter, with 317.8 times the mass of Earth, at a distance of 45,500 kilometers to have an orbital period of 90.16 minutes.
That would mean orbiting thousands of kilometers below the clouds of Jupiter.
And according to the orbital period calculator, the moon would have to orbit the center of a planet with 5 times the mass of Jupiter, or 1,589 times the mass of Earth, at a distance of 78,000 kilometers to have an orbital period of 90.9 minutes.
And according to the orbital period calculator, the moon would have to orbit the center of a planet with 10 times the mass of Jupiter, or 3,178 times the mass of Earth, at a distance of 98,000 kilometers to have an orbital period of 90.25 minutes.
And according to the orbital period calculator, the moon would have to orbit the center of a planet with 13 times the mass of Jupiter, or 4,131.4 times the mass of Earth, at a distance of 106,750 kilometers to have an orbital period of 90.0 minutes.
Most planets with mass between 1 Jupiter mass and 13 Jupiter masses would have radii in the range of about 57,200 to 85,800 kilometers.
And according to the orbital period calculator, the moon would have to orbit the center of a planet with 8 times the mass of Jupiter, or 2,542.4 times the mass of Earth, at a distance of 90,800 kilometers to have an orbital period of 89.99 minutes.
So your moon can orbit above the clouds of gas giant planets with a period of 90 minutes if those gas giant planet have masses of about 7 or 8 times the mass of Jupiter to 13 times the mass of Jupiter.
But there are other problems.
Here is a link to an article discussing the habitability of exomoons.
In it the authors, Rene Heller and Roy Barnes, introduce the concept of the habitable edge around a gas giant planet. tidal interactions between a planet and its moons can cause tidal heating in the moons.
If the tidal heating is strong it can make an otherwise too cold moon warm enough for liquid water and life. If tidal heating is strong it can make another wise warm enough moon too hot.
If the total energy per unit of surface area received from the star and from tidal heating exceeds a certain limit, more and more of the liquid water on the surface will become water vapor which is a strong greenhouse gas and will warm up the surface more. Eventually all the liquid surface water will become water vapor and the planet's surface will be dry and unable to support life. This is called a runaway greenhouse effect.
An even stronger internal heat from tidal heating can make a world a volcanic hell like Io, a moon of Jupiter.
Heller and Barnes's calculations indicate that for two specific exoplanets known at the time, satellites would have to be at least 10 planetary radii from the planet to be outside the habitable edge and thus safe from a runaway greenhouse effect.
In another article Rene Heller and Jorge Zuluaga discussed the magnetic fields of giant planets and their possible positive or negative effects on the habitability of exomoons that might or might not have their own magnetic fields.
Their abstract says:
Moons at distances between about 5 and 20 planetary radii from a giant planet can be habitable from an illumination and tidal heating point of view, but still the planetary magnetosphere would critically influence their habitability.
If planets more massive than Jupiter probably have radii between 57,200 and 85,000 kilometers, 5 to 20 planetary radii would be about 286,000 to 1,700,000 kilometers, and moons at those distances would have orbital periods far longer than 90 minutes.
So you should either chose an orbital period much longer than 90 minutes, or else make your habitable world orbit a brown dwarf.
A word orbiting a brown dwarf with 70 times the mass of Jupiter or 22,246 times the mass of Earth at a distance of 187,100 kilometers will have an orbital period of 90 minutes.
A word orbiting a brown dwarf with 75 times the mass of Jupiter or 23,835 times the mass of Earth at a distance of 191,500 kilometers will have an orbital period of 90.03 minutes.
So a world orbiting even the most massive brown dwarf on the edge of being a star with an orbital period of 90 minutes would be inside the habitable edge and thus far too hot.
Brown dwarfs probably emit a lot of infrared radiation, and so their habitable edges should be even more planetary radii out than for giant planet, making the problem even worse.
Thus I am forced to conclude that your habitable world needs an orbital period much longer than 90 minutes if it is a moon of a giant planet.
A moon with an orbital radius of about 350,000 kilometers around a planet with a mass of 13 Jupiters or 4,131.4 Earths would have an orbital period of 534.3 minutes or 8.9 hours or 0.37 days.
And I don't think that you will be able to get a much shorter orbital period while keeping a moon of a gas giant planet outside of the habitable edge.
So I suggest making your world a moon of a super Earth or mini Neptune as suggested above, which will have a much smaller radius and thus habitable edge.