NL25 AA makes nut full house

[rh2020]

My table image is very tight. I know I should cbet this don't remind me. Anyway, luckily for me villain likes his hand especially considering I was preflop raiser. He is running at 36/14/2 and is playing pretty poorly.

I made a pretty bad job of getting his stack, but anyway, questions below :-

a) ignore my flop check (although it is rainbow flop so he puts me on an ace if I bet right). do you check raise or call his bet? If you c-r, how much to?
b) my turn c-r is obv a bad move here, I figured he would fire again and originally planned to call but tried the c-r to bloat the pot but this indicated too much strength and the 9 is a bad card to c-r. Is the turn check worse than the flop check?
c) assume I check called the turn, how much do you lead out on the pot? 3/4?

No Limit Holdem Ring game
Blinds: $0.10/$0.25
9 players
Converter

Stack sizes:
UTG: $20.20
UTG+1: $24
MP1: $9.15
MP2: $56.65
MP3: $16.80
CO: $16.40
Button: $19.35
SB: $27.85
Hero: $18.50

Pre-flop: (9 players) Hero is BB with A[img]/images/graemlins/diamond.gif[/img] A[img]/images/graemlins/spade.gif[/img]
4 folds, MP3 calls, CO folds, Button calls, SB calls, <font color="#cc0000">Hero raises to $0.75</font>, MP3 calls, Button calls, SB folds.

Flop: 9[img]/images/graemlins/diamond.gif[/img] T[img]/images/graemlins/spade.gif[/img] A[img]/images/graemlins/heart.gif[/img] ($2.5, 3 players)
Hero checks, MP3 checks, <font color="#cc0000">Button bets $1</font>, Hero calls, MP3 folds.

Turn: 9[img]/images/graemlins/club.gif[/img] ($4.5, 2 players)
Hero checks, <font color="#cc0000">Button bets $2</font>, <font color="#cc0000">Hero raises to $4.5</font>, Button folds.
Uncalled bets: $2.5 returned to Hero.

Results:
Final pot: $8.5