How many mistakes in this hand?

[SmackeytheFrog]

In my first home game 1-2 NL foray, I am not sure whether I could have gotten away with this hand. Tell me where my own errors are.

#1 - bought in light. $300 max, I bought in $100. This led to a nit image (not inaccurate I think) where I dwindled to $35, sdoubling through the monster with QQ vs JJ, then again dwindling to a lucjky nut str8 flop against a 6 way preflop raise field, with 4 staying to river, so I'm up to $180.

Soon after getting some life, I wake up to AA UTG. I make standard raise at this table to $12. One call from cutoff who is a speculator and will sometimes go beyond flop with nothing, but not often. BB also calls.

Q44 flop. BB checks, I bet $25, cutoff calls, BB folds.

I do not make him for a lone 4, and if it's quads, so be it. I can reasonably put him on AQ, based on prior play.

Turn is Jack. I push with $142. He insta calls and slams down JJ. River no help.

#1 - did I underbet the flop?
#2 - since he showed interest on flop, should I check turn to see what he does?
#3 - why would I be so dumb as to buyin short?

[Check_The_Nuts]

I bet 30 on the flop. Pretty sure pot is $37. On turn the pots $87 so no don't overshove AA for $140. You said he calls with speculative stuff, how does he play after the flop after calling with this crap? Like if he has TT does he not put another penny in and just hope it goes check/check, does he bet the turn often after calling the flop? etc. etc.

Putting a player like this on exactly AQ is a huge mistake. He's got like QJ+ 55+ depending on how bad/good he is.