Have you stoved H2? I haven't, but it seems like this could be a call. It's $20 to win $35 with potentially 12 outs to the best hand, backdoor outs. If you can put T9 in his semi-bluff range than it's probably even closer.

H3 I am also betting the turn harder. The flush draw hitting is not really that relevant, and since we apparently don't have a read that the vill likes to play draws then I think he'll call $4.50 just as much as $3.75 with middle pair (99 type stuff) or a weak Ten. Check behind on the river is fine as many draws are now complete. If you had a better read there might be thin value there.