I go a bit higher on the flop, but half pot's not terrible. As played, I also go slightly higher on turn.
Turn options are wide open. Whatever you can do to get the most money in. I normally don't min-raise, but this isn't a terrible spot for it. Also, I'm not a fan of slow-playing, because a heart or diamond on the river is scary for both of us, and I'd hate to face a big river bet on a heart or diamond unless it fills us up (or better
).
So, click it back or shove -- whichever will work best vs. V.