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.
[1] 10^65 https://en.wikipedia.org/wiki/Timeline_of_the_far_future
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.
https://en.wikipedia.org/wiki/Roche_limit
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.
As for the spherical shape, the term there is hydrostatic equilibrium: https://en.wikipedia.org/wiki/Hydrostatic_equilibrium.
Keep in mind it's not actually possible to just "capture" an asteroid. The object needs to lose speed somehow in order to be captured.
Most likely via collision - something large hitting the planet, or two other moons crashing.
So the story of the moons is much more complicated than just "random piece of rock".
Pop culture references aside, yes, a moon is defined to be a natural satellite.
A fun read about this:
- [Asimov - The Relativity of Wrong](http://chem.tufts.edu/answersinscience/relativityofwrong.htm)
From the link above:
> 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.