Yes, perhaps these will be reclassified as satellites and moons will have to be large enough to form spheres under their own gravity. That would mean that Mars would have no moons though.
No, we're talking about a satellite forming into a sphere under its own gravity, not the planet's. Given a large enough lump of matter, gravity will overcome structural strength and force the material to flow into a roughly spherical shape. On Earth, this happens with the maximum mountain height being a few kilometers, but on Mars' moons, which are much smaller, the gravity is far too weak to overcome the rock's structure, and they are basically strange shaped rock things.
Mars' moons are really nothing more than captured asteroids. They look like "strange shaped rock things" because that's exactly what an asteroid is: just some random piece of rock.
Well, then you'd need to define an arbitrary time limit for how long it would take to form a sphere. In which case, you might as well have just defined a size/mass limit in the first place.
I think the definition would be that they're presently a sphere, that was formed under its gravity. That way objects could possibly become moons in the future, but the timescales are so long it's mostly irrelevant
That's technically correct (the best kind of correct!), but I think it's a misleading thing to state. The reason all objects eventually become spherical is because they eventually disintegrate into the nearest significant source of gravity, which will itself be a sphere. It's not as if, hypothetically speaking, an immortal human being floating through space without orbit would eventually morph into a sphere under the pressure of its own gravity. Rather any object is naturally going to decompose into parts which will be spherical or adjoined with an already spherical object.
Even if the conjecture that sufficiently high entropy causes quantum physical effects to dominate macro-physical spacetime is correct, long before that happens there won't be any objects left which aren't already spherical.
Actually, not always. A mass orbiting close enough to a planet is subject to tidal forces that will rip apart and eventually flatten a round body into a ring.
If body is in a strange orbit taking it in and out of the roche limit, it will adopt some strange middle ground between bring a round moon and a flat ring. Inside the limit, material will be ripped away from the body and flattened towards a ring. Outside the limit, that material will fall back nearer to the equator. So you get something like a ball with a belt.
According to quite a lot of comparative planetary geology work, Mt. Everest, Mauna Kea, etc. are pretty close to the height limit imposed by gravity and structural strength. On some objects (like Mercury or the Moon: see https://arxiv.org/pdf/1511.04297.pdf) there aren't any mountains high enough to hit that limit, but Earth seems to have active enough tectonics to push up at that limit despite weathering.
A bit of one, a bit of the other, AFAIK. Of course, having geological weathering is also a function of gravity; small objects don't have the atmosphere for it.
The tallest planetary mountain in the solar system is Olympus Mons on Mars, about 2.5x as tall as Mount Everest (as measured from sea level). This is only possible because Mars has 1/3 the gravity of Earth.
> To put it another way, on a flat surface, curvature is 0 per mile everywhere. On the earth's spherical surface, curvature is 0.000126 per mile everywhere (or 8 inches per mile). On the earth's oblate spheroidal surface, the curvature varies from 7.973 inches to the mile to 8.027 inches to the mile.
> The correction in going from spherical to oblate spheroidal is much smaller than going from flat to spherical. Therefore, although the notion of the earth as a sphere is wrong, strictly speaking, it is not as wrong as the notion of the earth as flat.