dont mind the flop call, hoping someone raises behind or catches up.
I think you should raise the turn if for no other reason than metagame. If you had air, you want to fold out minbetter, so just raise it to a real bet amount.

also he leads flop weak (1/4th pot) then turn even weaker... he either has absolutely nothing, in which case you are going to win exactly his $1 river bet, or he's fishing for a raise.