$2 NLHE 6-max: TPGK OOP turn/river decision on dynamic board

888Poker Snap, Hold'em No Limit - $0.01/$0.02 - 6 players
Hand delivered by Upswing Poker

UTG: $1.00 (50 bb)
MP: $1.29 (65 bb)
CO (Hero): $1.54 (77 bb)
BU: $1.22 (61 bb)
SB: $1.18 (59 bb)
BB: $1.60 (80 bb)

Pre-Flop: ($0.03) Hero is CO with A♦ Q♥
2 players fold, Hero raises to $0.06, BTN calls $0.06, 2 players fold

Flop: ($0.15) 8♥ A♠ 7♥ (2 players)
Hero bets $0.09, BTN calls $0.09

Turn: ($0.33) 4♣ (2 players)
Hero bets $0.16, BTN calls $0.16

I think I should bet bigger on the turn and fold to raise vs unknown.

River: ($0.65) 9♣ (2 players)
Hero checks, BTN bets $0.91 (all-in), Hero ??

Not sure which line is better here bet-fold , check-call or check-fold.

Villain is an unknown, default max buyin for NL2 6max on 888 poker is 50BB

Questions:
1. How would you play this hand?
2. What do you think villain have?

In a low stacks, on-line game Villain's range is wide. 66+ Broadways, suited connectors, suited aces. The bet preflop and on the flop are standard and only define Villain's hand a little. I don't think betting more on the turn would get Villain to fold, given that she pushed the river.

Hero is losing to 88, 77, and 99. I doubt 44 would have called the flop. Hero is also losing to AK, A8, A7, A9 and A4, 87 and 89. I doubt Villain has 65. or JT for straights. AA, KK or QQ, she would have bet earlier. It looks to me as if Villain has trips, JJ, TT, or an Ace.

Hero is beating AJ, AT, A6, A5, A3, & A2 all of which I would suspect she would check to see a show down with the weak aces. Given the number of combos that she is beating Hero with, I would fold to the river shove.

I'm guessing she had trips or a suited ace.

I would give button value hands {A9, TJ, 99} not that many given he might overvalue Ax hands + bluff any heart (any) and some other draws. Most of the time he would raise turn with sets. I would play the hand exactly like hero did and cry call river (to lose TJ I’m quite sure = because it hurts the most sigh).

What an awful spot. Very hard decision OOP.
I think the way you played is fine. There is merit to checking the turn OOP to protect your checking range, but betting is also fine.
Definitely check river, look to check call a bet less than pot size. When he shoves it's really tough there are a lot of hands that beat us. The 9 is an awful card for us because the straight draws got there and villain now has 2 pair combos that beat us.
I'm leaning towards a fold here.

1)
Apart from sizing would play similar. Betting out on the river does not make a lot of sense to me so I would prob check fold (on a huge bet) need at least AK/AQ without hearts in it to call a huge river bet (gives him more missed flush draws) this makes that you are a bit balanced between calling and folding on the river and difficult to exploit

2) Opponents range:
Sets:
77, 88
Straights:
JTh other straights should have folded on either flop or turn
Two pair:
A7, A8, A9 all suited might even have 87s & 89s but very unlikely, but would they all bet the river?
Top pair hands:
AK, AQ, would they bet the river? and AK and even AQ might 3 bet preflop
Missed flush draws hands
KJh, T9h, J9h, KTh (will they all call both preflop and on turn?)
Missed straight draws,
T9s, TJc, T9d (will they all call preflop and also on turn?)

Villain had JTo here and agreed I should be folding to the river shove as there are more value hands than bluff