Hand #1:

Ace-garbage off sucks in this position (and most other positions BTW) so you should toss it like it is burning your fingers. There will likely be some here that will defend the half bet call but I don't think it (even getting 7:1) could ever be profitable in the long term. There are times you may try to snap off a steal but that's not what is happening here.

This flop illustrates the difficulties of playing such a hand. The problem is magnified by your position. The bet is okay if the thought was to steal the pot. Even though Ace-garbage is a bad hand to play with you will see many, many micro players who always limp with it preflop. Here you have to fear (and are behind) A9, A8, A7, and A3 as well as "legitimate" Ace holdings like AJ, AT and (incorrectly) limped AK and AQ. You are ahead of only A2, A4 and A5 as well as any small pairs (except pocket 9s or 3s). The pot is small (4 SB). Your choices are to bet out and hope everyone folds (not likely with four in but it does happen), to check/ fold, or to check/ call (don't bother). So you bet out and get one taker, unlucky, get ready to fold the turn to any strength.

On the turn, nice try (again), but you I think you have to fold to the raise.

Hand # 2:

All looks good. WTF is MP1 check raising the turn with? A set? It has to be. Against an unknown or a known good player I would respect this raise. Against a LAG or a SLAG I would probably 3-bet, but this is probably not correct. I spew a lot with TPBK against a probable set. Slowing it down as you did is probably correct.