Its my understanding you can use Power Attack with Attacks of Opportunity. If so, then feats that feed off Power Attack also in effect? For example, I use my AoO and hit. I can choose to lose Power Attack bonus and instead push with Pushing Assault? If yes, what happens to the movement of the person I hit, they just start moving again from whatever spot they're in?
Yes you can power attack during an AoO and you can use your feat to push them. By subbing out the extra damage.
You are correct, if your target drew an AoO and you pushed them the nthey would still be able to move (if they had any distance remaining) as you did not impede their movement in any way other than the push. That is unless the target took a five foot step and you readied an attack, pushing them away and out of melee range. Or you pushed them into difficult terrain of some kind.
If you want to stop them then check out the feat Stand Still or the trip combat maneuver