I note that astronomical bodies are requested with diameters and distances that give them angular diameters of about 0.5 degrees. An object will have an angular diameter of about 0.5 degrees when it is at distance of about 114.59165 times it's diameter.
The first thing that Overlord - Reinstate Monica should realize is:
The length of seasonal cycles on the Earth sized moon will depend on the orbital period of the planet around the star, not the orbital period of the Earth sized moon around the planet.
Overlord - Reinstate Monica wrote:
The Earth-like moon has a cycle of seasons lasting 360 real-life days. This does not necessarily mean it orbits the gas giant in 360 days, it just means that winter, spring, summer, autumn last about 360 real-life days. (Earth-like seasons)
The sub-satellite orbits the Earth-like moon in about 30 real-life days.
In real life a giant planet will exert tidal forces upon all its regular satellites - but not captured asteroids in distant orbits - that would adjust the satellite orbits so that the regular satellites, including hypothetical giant Earth like satellites, will have almost circular orbits in the equatorial plane of the giant planet. The tidal forces will also tidally lock the rotation of all the satellites so that their rotation periods or "days" will be the same length as their orbital periods around the giant planet or "months". Thus one half of each satellite will constantly face the giant planet and the other half of each satellite will constantly face away from the giant planet.
And the giant planet will do this very fast by astronomical standards, in a few tens of millions of Earth years. Since seasons are caused by the axial tilt of an astronomical body relative to its orbit around its star, and since the Earth like moon must have the same axial til as the giant planet, the cycle of seasons on the Earth like world will have the same length as the orbital period of the giant planet around its star, not the orbital period of the Earth like moon around the giant planet.
An orbital period of 360 "real days", presumably Earth days, of the giant planet around its star and within the habitable zone of that star is certainly possible.
The shortest known year of an exoplanet in the habitable zone of its star is 4.05 Earth days, for TRAPPIST-1d, while Earth is in the habitable zone of the Sun and has a year 365.25 Earth days long, and planets in the outer limits of the habitable zones of some stars should have years even longer than Earth does.
Our solar system has many examples of natural satellites of planets, dwarf planets, asteroids, etc. But there are no examples of any known natural sub satellites orbiting natural satellites in our solar system. There are no known natural satellites, or exomoons, in other solar systems, though there are a few candidates. If astronomers are not certain they have detected any exomoons yet, they certainly could not have detected any sub satellites in other solar systems.
And I believe that there are calculations of the orbital stability of sub satellites that indicate that natural sub satellites would be very rare and there would probably thus be a very restricted set of stable orbits that a sub satellite could have to orbit any particular specific satellite. Thus it would probably be very unlikely for a natural sub satellite to have a stable orbit in your fictional star system where you want it to be.
Assuming for the moment that there actually are a lot of sub satellites in other star systems and that our solar system is rare in not having any, we can make some assumptions about the orbits of your planet, your Earth like moon, and your sub satellite.
Includes the statement:
The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a)
So if your planet orbits its star with a period of 360 days, your Earth like moon would have to orbit the planet with an orbital period equal to or less than one ninth or 0.11111 of the planet's year, or 40 days or less.
Assume for the moment that the orbital period of the sub satellite of the Earth like moon must have a similar relationship to that of the Earth like moon to that of the planet, in order to have long term stability. In that case the orbital period of the sub satellite round the Earth like moon would have to be one eighty oneth or less of the orbital period of the planet around the star, or 0.0123456 or less. With an orbital period of the planet around the star of 360 days, the orbital period of the Earth like moon around the planet would be 40 days or less and the orbital period of the sub satellite around the Earth like would be 4.444416 days or less.
Assume that the sub satellite has an orbital period of the desired 30 day length. Then the orbital period of the Earth like moon around the planet would have to be at least 270 days and the orbital period of the planet around the star would have to be at least 2,430 days.
Assuming that you meant that the Earth like moon would orbit the planet with a period of 360 days, and the sub satellite would orbit the Earth like moon with a period of 30 days. In that case the Earth like moon would have an orbital period twelve times as long as the orbital period of the sub satellite, and that seems sufficient to me. Then the planet would need to have an orbital period at least nine times as long as 360 days, or at least 3,240 days.
I suggest that you may need to give up the idea of having your Earth like world a moon of a giant planet and also have a sub satellite orbiting it. There would be no problems with making your Earth like world a satellite of a gas giant planet if your Earth like world didn't have a sub satellite orbiting it. And there would be no problems with making your Earth like world have a satellite orbiting it if the Earth like world was not itself a satellite of a giant planet.
But of course such a situation would result on only one astronomical body being able to eclipse the star of your Earth like world.
Or would it?
If your Earth like world is an Earth sized moon of a giant planet, maybe there are one or more other large moons orbiting that giant planet. And maybe one or more of the other large moons will sometimes be in position to eclipse the star of the system as seen from the Earth like moon.
Or possibly your Earth like world would be a planet orbiting its star. Such a planet could have no moon at all, like Venus, one moon, like Earth, two moons, like Mars, or possibly three or more moons. Of course the relative orbits of the moons would have to be such that they do not destabilize each other.
The Moon orbits the Earth at an average distance of about 384,399 kilometers. The Moon has an average radius of 1,737.4 kilometers & average diameter of 3,474.8 kilometers. At it's orbital distances, the Moon has an angular diameter of 29 minutes & 20 seconds to 34 minutes & 6 seconds, a range that includes the 0.5 degrees asked for.
If the moon was at a distance of about 192,200 kilometers, half of the distance in real life, it could have a diameter of about 1,737.4 kilometers and an angular diameter of about 0.5 degrees.
If the Moon was at a distance of about 96,100 kilometers, a quarter of the distance in real life, it could have a diameter of about 868.7 kilometers and an angular diameter of about 0.5 degrees.
If the Moon was at a distance of about 48,050 kilometers, an eighth of the distance in real life, it could have a diameter of about 434.35 kilometers and an angular diameter of about 0.5 degrees.
If the Moon was at a distance of about 768,798 kilometers, twice as far as in real life, it could have a diameter of about 6,949.6 kilometers and an angular diameter of about 0.5 degrees.
So it seems reasonable to suppose that the Earth could have two or more moons orbiting it in stable orbits at various distances that might all have angular diameters of about 0.5 degrees.
And there is this article https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/2 at the PlanetPlanet blog, that cites a scientific study showing that seven to forty two bodies with the same mass and equally spaced can share the same orbit with long tern stability.
So possibly a giant planet could have a ring of equally spaced Earth like moons around it, or an Earth sized planet could have a ring of moons around it.
Earth has a diameter of about 12,742 kilometers, so it should have an angular diameter of about 0.5 degrees at a distance of about 1,460,126.8 kilometers.
So if there could be seven to forty two equally space Earth sized moons orbiting a giant planet, the total circumference of their common orbit would be about 10,220,887 to 61,325,325 kilometers, thus giving the radius of their common orbit about the giant planet at about 1,626,706 to 9,760,236.6 kilometers.
And each of those seven to forty two Earth sized moons orbiting the giant planet might sometimes see the star eclipsed by one or another of the other Earth sized moons, and also sometimes see the star eclipsed by the giant planet.
The giant planets in our solar system have average diameters ranging from 49,532 kilometers (Neptune) to 142,984 kilometers (Jupiter). So they would have angular diameters of about 0.5 degrees at distances ranging from about 5,675,953.6 to 16,384,772 kilometers.
Combining the two calculations, a giant planet might have a ring of twenty five to forty two Earth like moons in a shared orbit with a radius of 5,809,664.8 to 9,760,236.6 kilometers. The two closest Earth sized moons to another would have an angular diameter of about 0.5 degrees, and if the planet had a diameter between about 50,698.8 kilometers and about 85,174.064 kilometers it would have an angular diameter of about 0.5 degrees in the sky of each of those Earth like moons.
I haven't calculated how long the orbital periods of the Earth like Moons at those distances would be.
And an Earth like planet could possibly have a ring of seven to forty two moons with distances about 114.59165 times their diameters, so that they appeared to have angular diameters of about 0.5 degrees.
Since such a ring of moons could be at most only a few times as far from the planet as the Moon is from Earth, the planet could have eclipses of its star by one or another moon several times as often as Earth does.
Note that the PlanetPlanet article more or less assumes that such a ring of equally spaced and equal mass objects orbiting a larger object would be extremely unlikely to occur naturally and would probably have been constructed by a highly advanced civilization.