I think you should have raised on the flop on hand 1. Personally, I prefer to raise when I'm assured to have the best hand and when there is some risk that I won't on the next card, especially when another opponent seems confident in his holdings as well. Just calling with 2 diamonds on the board is a bad idea for a couple reasons. Not only could someone draw to a flush, but a third diamond could kill your action if no one does. So, I think you should make some kind of a raise on the flop, and then on the turn make a large bet or raise (unless the board pairs, in which case you may want to find some creative means of getting your opponents all-in), all-in if necessary. You only want this to go to the river if the board pairs, or if one or more opponents are all-in before then.

On hand two, your opponent probably has either a 6 or a flush on the turn, and a raise will help gauge his commitment to the hand, as well as build the pot. You may get reraised, and then you have the option either to reraise or just call and see how your opponent behaves on the river. The strength your opponent shows on the river, I think, should determine the amount that you bet or raise.

Also, I think there is nothing wrong with calling a small bet on the flop in hand 2. Though it's unlikely, you may actually have the best hand, and if your opponent has a 6 or flush draw, you can win an enormous pot by hitting a 7 as you did.