[schwza]

1: that's a tough hand. i think pretty much any line you could take (check/fold, c/r, check/call, bet/call, bet/fold) would not be too bad.

i think checking is best. if there's a bet and a raise behind you, you can fold comfortably. if a relatively short stack bets, you can c/r a-i. if you check and someone with a comparable stack bets, i'd fold. it's going to be impossible to extract a lot more if you're ahead, and you're at risk of getting moved off the best hand if you don't improve. also, villain is guaranteed to have 6 outs. and depending on where the bet comes from, you might be risking a c/r behind you.

calling a small bet would be good though.

a couple questions: do you have a diamond? what are rough stack sizes of villains?

*******
2:
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I figure he could have the following hands: AsKs/AsQs, AJ, or a set.

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if that's the range that you think he has, it's an easy fold. i think the only way you can call this is if he often has hands like AxQx or 44 and his plan was to call the flop and push if you only fired one continuation bet. unless you have that good a read, you've gotta let it go.