Very first hand, BB is a complete unknown.

Is it a good spot to stack-off? Vs competent villain I don't think I'm ever ahead unless he's a fish.

What do you think his range is?
Do you ever fold here given the nice pot odds?

888 Poker - $10+$0.60|10/20 NL (6 max) - Holdem - 6 players

MP: 500.00
CO: 500.00
BTN: 500.00
Hero (SB): 500.00
BB: 500.00
UTG: 500.00

Hero posts SB 10.00, BB posts BB 20.00

Pre Flop: (pot: 30.00) Hero has A Q

fold, MP calls 20.00, fold, BTN raises to 80.00, Hero raises to 240.00, BB raises to 500.00, fold, fold, Hero calls 260.00

Looks standard to me. You're definitely committed after your 3bet.

have a plan the next time you put money in the pot

So, what do you do? Do you flat preflop OOP instead? Fold AQ? 3-bet less/fold to a shove?

with 10/20 blinds a person who makes it 80 pre is almost never folding. So if you think you're ahead of his range then shove for value, if not then fold, not much room to do anything else. In this case I'd say it's a snap shove, and do a dance when he calls with a4. Flatting AQ to a 4x BN raise in a ST is terrible.

What would you guys do if we were the BU and raise to like 55 with AQ and an unknown 3bets to 140? I guess its not solvable without reads so it doesnt really matter what we do or what do you think?

Considering the stacks and pot already, and your likelihood of being ahead of BTN, this should have been a shove preflop instead of 3bet.

I tend to agree. The only merit of smaller 3-bet is to get hands like A9-AJ to call or shove. But that probably isn't worth it.