Bottom Set - did I overplay or underplay my hand?

[antifish225]

20-40 full B&M game - table is mainly TAG/solid players - most capable of making good/tricky moves in moderation - hand in question:

UTG+1 limps, MP1 raises (he has been raising allot in this session,however he has been catching very nice cards), 2 late callers, button folds, I am in SB with 44 and call, BB folds - 5 players see the flop ($220 pot).

Flop A Q 4 rainbow (giving me bottom set)

I bet, early limper calls, MP1 raises, 1 late position caller - (at this point I am putting him on AK or AQ) - I reraise - limper folds - MP1 calls - late position calls.

Turn: 8 (no apparent help)

I bet, MP1 raiases, late player folds - I ??? (thought about re-raising, but flat called instead as my read now was either AK, AQ, QQ or AA)

River 6 (again no apparent help)

I check, MP1 Bets, I ???? (think about raising but flat call)......

He flips over AA for top set - beating my set of 4's....

So here are my questions:
Did I underplay this hand (given the outcome I don't think so - however I am not really concerned about individual outcomes - more playing correctly)
Did I overplay this hand?, could this be laid down on the turn raise for a long term +ev?
Would you play it differently?

Your thoughts are appreciated.........(btw I think I under played my hand on the turn....)