Both Mars’ moons (Phobos and Deimos) are asteroids. They’re really small.
“Hydrostatic equilibrium” is the term (I think) for when a body is massive enough for it’s gravity to pull it into a spheroid. E.g. Ceres is about 950km in diameter and is ‘round’.
Probably. It does happen.
Fun fact. Triton, Neptune’s largest moon, orbits the wrong way which means it was gravitationally captured by Neptune at some point.