I don't think I cbet this. You're basically turning a hand with SD-value into a bluff right on the flop.
On the turn, you're basically barreling against 55-66-77 only, which is a small part of his range. If you want to do this, you should probably make it $2.50 and be prepared to fire the river again (although this card is obviously not the greatest).
I'm cbetting the flop just for protect my hand from overcards, on the ace i try to make fold even some hand like 54s 64s maybe K4s and ofc 33 55 66 77. His fold to cbet is 57% (on 9 samples) so it seems realistic to me.
it does look like a pocket... and if he has one, hes calling turn even if A shows up... i guess you should 3barrel then... i'd give him TT depend how fast hes calling
once he c/c turn, he often also has Ax himself, so I think river check is fine, although likely you´re not ahead there often,
Player needs to be quite calling station to c/c turn with medium pp, so I think you often get looked up on the river