First hand has a slight donkey odor to it… Raising in that spot with AK is right, but your raise is pretty week. What will that almost min-raise accomplish? Most often, it will invite multiple callers, but worse, you have gained little to no info about what you’re up against. They could have any two suited or connected cards, any ace… You’re probably not up against AA-JJ, or you would have probably seen a reraise, but other than that, you’re in the dark. So when the flop hits, the ace is good, but you may welll be up against two pair or a straight draw.
Once villain calls your raiseon the flop, he’s announcing he has a hand, so when he checks on the turn, I think it’s fine to check it as well. There’s that simple adage, "big pots are for big hands"… and all you have is top pair here. So don’t commit everything here on that one pair vs. an opponent who seems to be indicating strength. Unless you have some great read that this guy is a fool who over-values weak aces, I think checking the turn would make more sense here.
On the second hand, you’re shortstacked at this point, and a suited ace may be as good as you get for a while. I think a push here is fine, but you’re often going to be a coin-flip at best once someone calls…. This hand really isn’t a good example to measure your decision-making, since once you get to this shortstacked state in a tourney, your decisions are a lot more simple and less strategic. The hands to analyze, I’d suggest, are the ones that got you to this point…