No reads.

PokerStars No-Limit Hold'em, $0.02 BB (8 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

Hero ($2.97)
MP1 ($2.87)
MP2 ($0.89)
CO ($3)
Button ($2.28)
SB ($0.87)
BB ($2.67)
UTG ($1.31)

Preflop: Hero is UTG+1 with 4, 4. CO posts a blind of $0.02.
1 fold, Hero calls $0.02, 1 fold, MP2 calls $0.02, CO (poster) checks, Button calls $0.02, 1 fold, BB checks.

Flop: ($0.11) 4, K, 3 (5 players)
BB checks, Hero bets $0.11, MP2 folds, CO folds, Button calls $0.11, BB folds.

Turn: ($0.33) 7 (2 players)
Hero bets $0.3, Button calls $0.30.

My thinking: 7h is a bad card as it completes a flush draw and a straight draw. I bet because his calling range on flop > {flush draw + straight draw}, I have 10 outs if he's outdrawn me, and I get no information by checking. How is my bet size? Should I have bet 1/2 or 2/3 pot instead of full pot? Does it even matter?

River: ($0.93) A (2 players)
Hero checks, Button bets $0.42, Hero calls $0.42.

My thinking: I briefly considered b/f and opted for c/c because he doesnt fold any better hands and I feel my hand still had some showdown value. Thoughts? Should I have considered c/f? Or am I weak-tight for not wanting to felt with a set at 2NL?

Any comments on how I played the hand would be greatly appreciated.