consider some xbacks on F with this hand...( i see my self checking here a lot )
As played time to fold on river for sure .. i dont even see a reason to spaz with worst hand here and posisitons.if u owned him nh .. he prop was tilted :D

Is it the notation or did we raise 4x UTG? I'm going on that assumption with my analysis.

So he sees you open to 4x UTG and should put you on a tighter (strong) range. We get called from the BB and he has to pay 0.75 to win 2.10 so he doesn't get that great of immediate odds. So he either has a strong hand himself that he's slowplaying or one with implied odds. So something like lower pockets, suited Ax(although I don't like the call with those as often), strong Ax.

Flop comes A high and we cbet. The story you are telling is 'I have an A or better.'. We get raised on a dry flop. I don't mind calling perse but BB is saying 'I beat an Ace'. We call ('I don't believe you' or 'I have better than an ace').

So far we have shown strength preflop (UTG open to 4x) and strength on a dry flop (cbet, call raise). BB then barrels off into us and we call it down.
I don't mind perse, I've seen people do this with total junk but they had much different stats than BB in this hand (although 3-bet of 1%...). I just don't think this is profitable against their range.

You also ask if it's always a fold on turn. The answer is that you should have decided that on the flop. To me this is a fold on flop. Yes, it can be exploited. No, almost no one is doing that at 25NL.

As Kafrinius said, I actually like a check back on flop. We flopped good for our exact hand but it's hard for villain to be calling three streets with worse here. Underpairs call 1 street, maybe 2. Weak Ax call 1 to 3 streets, mostly 2 (fish call 3 streets). But even if he'd call 3 streets, we're only exploiting a small part of his range.
A check back allows us to control the potsize in position and it induces much more bluffs and weaker calls from BB.
Imagine he has any pair from 66 to KK. You cbet flop, he calls. Turn you bet, he folds a large %. Now check back flop instead. He's not believing when you bet turn and calls. River comes, we bet and he calls more often than before. To get that extra % when he calls we give him 2 outs.
Imagine he has 78s+ (excluding Ax). He folds flop close to 100%. But with a checkback he might bluff turn by leading. Extra street of value. Very little equity given (sometimes gutshot, some backdoors (4%)).
If he has AJ-A6. We cbet, he calls. We bet turn, he calls. We bet river, he folds often, calls sometimes. If we check flop he either leads or check calls turn. And continues river or check calls. Two streets most of the time. No equity change as we always get to river.

There's also the possibilty we are behind. By checking back we keep the pot smaller and lose less. If he has a set he bets turn, we call, he bets river, we call. Even if he bets pot twice that's a loss of 8.40 compared to an entire stack.
On top of that, we almost always get to showdown by checking back flop. This can't be underestimated. By checking back flop if becomes much harder for villain to get us off our hand.
And to finish, do you cbet JJ+ here? Do you cbet 66-TT here? I most often won't. I would want to get to showdown with them as cheap as possible. So checking back Ax makes your opponent believe you have one of these hands and bluff too often.

In short: Fold flop as played. In the future check back this hand on such a dry flop.

Your hand is basically a bluff catcher at this point you said you felt he was aggro and that's why you called off. If you thought he was bluffing your play makes sense. I don't know anything about him but it really looks to me like he had A4, 55, 54, A3, or total air. Would love to know if it's any of those.

With the 1% 3-bet and supper high AF, I wouldn't be surprised to see AK+. His range will have a ton of two pair combos as well OTF. He will also have some 76s, 87s, 22/33 ( maybe) for bluffs, maybe some hands like TP+GS, maybe some button mashes. Wouldn't bet/fold Top2nd quite yet.
Turn is a good card for us only improve his value hands...
The river improves 22,33,A2s,A3s (if he had those hands) giving him even more value hands...
I think we could give him some bluffs like 87s and 67s and it would still be a fold OTR. Probably good like 20-25% of the time here against a really aggressive opponent.

I don't think betting the flop is the worst thing in the world, although checking has allot of merit as others have mentioned....

Yes, AK is going to be a big part of his value range here. 1% 3bet, UTG opens 4x. I'd be shocked if this player 3bets AK in this spot.

As for checking back flop, yes it has merits but we can't be checking back all our Ax here and we need to have every hand strenght in our range for every line that we take.

So we need to have an Ace here when we bet 3 streets, when we check back flop and bet T/R and when we bet/check/back and also when we check back flop and turn

So, My default gameplan in this spot (UTG v BB - dry Axx) will be to bet 3 streets with AK, check back A2-A5s and mix up how I play ATs+ AJo+ between Bet/check/Bet - Check back/Call/Call or Check back/Bet/bet

AK should probably always be a bit otherwise what other value hands can we bet 3 streets with ?

Against this player, given our 4x utg open and BBs tight 3b range AQ would mostly likely gointo my flop check range or my bet/check/call range