Sorry I cannot begin to explain a single one of those bets on any street. Player 2 limps pre in bad position with a questionable hand and gets min raised by the BB who holds 34o? I don't get either of those. I do call the min raise with JQ though... I guess that is one part that I do understand. On the flop the min bet is probably just to stop the action cheap with a drawing hand and hopes to hit a 5 but it doesn't work, is raised and he calls his draw anyhow. The BB checks now not hitting his draw and the JQ fires a second shot and gets called. I guess this is an ok play as your opponent has not really shown any strength by min betting the flop and checking the turn so he continues to represent an Ace. Check down on the river.
That's my best explanation.