RE here. I believe I may help.
First, welcome!
Second, read their Wiki, it answers most of your questions: wiki.tfes.org
I've heard the general consensus is that the sun is a sphere, but acts as a spotlight. There's a few random FEs around the internet that claim its flat, but not many.
FE come to the conclusion that its flat, as based off what I've read and watched, because they have claimed to have never detect curvature, spin, and have only ever seen a flat horizon.
Historically, it popped up before the notion the earth is round came about (2000+ yrs), but went into serious decline after the model of a round earth take over:
https://en.wikipedia.org/wiki/Flat_Earth#HistoryThe phases of the moon wouldn't be determined by the Earth. Even on a round earth, the Earth doesn't cause the phases, rather its position relative to the Earth. The changing angle of sunlight bouncing off the moon caused by its orbit is what determines the phases. As for flat earthers, the wiki (
https://wiki.tfes.org/The_Phases_of_the_Moon) uses the idea of sunlight reflecting off the moon, except that the change in the phases is caused by the change in altitude and position of the sun and moon over Earth. Here's their explanation:
When the moon and sun are at the same altitude one half of the lunar surface is illuminated and pointing towards the sun, This is called the First Quarter Moon. When the observer looks up he will see a shadow cutting the moon in half. The boundary between the illuminated and unilluminated hemispheres is called the terminator.
When the moon is below the sun's altitude the moon is dark and a New Moon occurs.
When the moon is above the altitude of the sun the moon is fully lit and a Full Moon occurs.
The time between two full moons, or between successive occurrences of the same phase, is about 29.53 days (29 days, 12 hours, 44 minutes) on average. This denotes the cycle of alternating altitudes.
Flat Earthers (again, from what I've usually seen them claim, this may not be correct for everyone) usually say the curve seen from a plane isn't actually there, and you would need to be higher to see it (which is partially true on a round earth. Curvature on a RE would be very slight from that altitude, you'd need to roughly double it to see anything significant, like riding Concorde), or the windows on the plane cause a fisheye effect, etc. On the wiki, it is claimed curvature is just the terminator line from the spotlight:
https://wiki.tfes.org/High_Altitude_PhotographsTo reiterate, these are based off what I read online or see in FE videos, this may not be the case for everyone, but this is, I assume, the majority of belief. Of course, there's always other opinions.