1) 4x4x1.5 = 21cu/in = 9- #12 conductors.
ALL the grounds count as one, so you can have four 12/2 cables in this box.
For #14 you can have 10- #14 conductors, which is also four 14/2's, or three 14/2 and one 14/3.
2) It's ok, but not ok IMO in your situation. I'd say it's fine if there were a decent gap, like an inch. Just enough to slip a cable in is not enough to prevent damage to the cable.
3) Yes. It is marked in the box or on the cover.
4) Yes, provided you are not switching more than 15 amps (highly unlikely in a home).
Sometimes I feel like if I answer any more questions it is like someone trying to climb over a fence to jump off a bridge and me giving them a boost.