I would like to better understand how often solar eclipses would happen on a planet that is tidally locked with its moon (and the moon is tidally locked with the planet). I understand that in this situation, from the planet’s surface, only half of the planet would ever see the moon and thus ever be subjected to a solar eclipse. I also am under the impression that, from the planet’s surface, the side that could see the moon would witness the full range of lunar phases over a day. Assuming in this situation that the moon orbits the equator of the planet, and the planet sits on a rotational axis similar to the Earth’s (23.5 degrees), would the hemisphere of the planet that sees the moon experience only a solar eclipse twice a year – on the Spring Equinox and the Autumnal Equinox? If so, how much of that hemisphere would be able to view a total and partial eclipse? What factors would influence the total and partial eclipse? Thank you!
Assuming in this situation that the moon orbits the equator of the planet, and the planet sits on a rotational axis similar to the Earth’s (23.5 degrees), would the hemisphere of the planet that sees the moon experience a solar eclipse twice a year – on the Spring Equinox and the Autumnal Equinox?
Not necessarily. If the planet has any inclination at all, then the line towards the sun will fall in the plane of the moon's orbit for only an instant. You are only guaranteed to have an eclipse if the moon is in the right spot at precisely the instant of the equinox. If the moon is on the wrong side of the planet at that time, the planet-moon system may have moved too far for an eclipse to occur by the time it orbits back around.
How close the line-up has to be depends on the relative angular sizes of the moon and sun in the planet's sky. If the year is an exact integer multiple of the length of the day, then any particular spot of the equator will either always see an eclipse, or never see an eclipse. If it isn't, then some years will have eclipses and some won't, with the frequency depending on how close of an alignment you need--which, again, depends on the relative angular sizes of the eclipsing bodies. A moon that appears significantly larger than the sun, and which moves quickly across the background stars, will produce eclipses more often than a smaller moon or a slower rotational period. With a sufficiently large moon and a sufficiently short day, you might even get multiple eclipses every equinox.
How large of an area is covered by a total or partial eclipse also depends on the relative angular sizes of the eclipsing bodies, and on the ratio of distances from the planet to the moon, and planet to the sun. A small sun farther away will produce larger umbras (total eclipses) and smaller penumbras (partial eclipses), while a larger sun closer to the planet will produce smaller umbras and larger penumbras.
Your assumptions are mostly correct. Although if the planets axis has a 23.5 degree tilt then some other parts of the planet would be able to see the moon on a seasonal basis, in a similar way that some stars are only visible a certain times of the year on earth.
The key parameters are the distance of the moon from the planet and the size of the moon. If the moon is too small or too distant then no eclipses would occur.
It is also important that the moon orbits the planet in or near the plane of the ecliptic. If this is not true then solar eclipses would be much reduced and would only occur near the equinoxes.
If the orbit was not circular but eccentric then the instances of eclipses would also be significantly affected depending on the nature of the orbit and complex to calculate (as is the case with earth’s moon).
Assuming the moon’s orbit is circular and in the plane of the ecliptic, the planet would experience a solar eclipse every day provided the moon was large and close enough.
The size of the umbra and penumbra shadows would depend on the size of the moon and its distance from the planet. A very large moon in a low orbit would create an eclipse that would almost cover the entire planet. Progressively smaller or more distant moons would produce smaller areas of shadow limited to the tropics which would cover a smaller area and be more fleeting. Distant or tiny moons might only produce partial eclipses and very distant or tiny moons would not produce eclipses at all their effect being more like that of the transit of Venus.