1). 3bet size is fine, as played i would c/f the river. Unless you are going to be going all in on the turn for value, betting the flop here with Q-Q is worse than checking.

2). Interesting spot. I guess it depends on how lightly you think these players will get in on the flop with that determines whether check/folding or bet/calling is best. Bluffing in 4 way pots (particularly reraised ones) is pretty much always bad play.

3). Call the flop, fold to a turn bet. By raising the flop you are turning your hand that has showdown value into a bluff. River shove seems best once you have raised the flop, gotten called, and been checked to on two streets.