I have a few questions about this hand. My first is, after I 4-bet pre, is this a flop where I should be range betting OOP? I was thinking about check/folding flop, but I thought it was ok to x/c with the bdfd. As played, I am wondering if x/c flop and x/f turn was the worst line (or at least worse than betting myself or check/folding). I only had 8 hands on opponent when the hand was played so they were basically unknown. Also wanted to ask, is the sizing ok preflop?

Winning Poker, Hold'em No Limit - $0.05/$0.10 - 6 players

UTG: $13.55 (136 bb)
MP: $13.78 (138 bb)
CO: $11.30 (113 bb)
BU: $15.07 (151 bb)
SB: $26.46 (265 bb)
BB (Hero): $10.17 (102 bb)

Pre-Flop: ($0.15) Hero is BB with K♠ K♣
2 players fold, CO raises to $0.25, BTN 3-bets to $0.70, 1 fold, Hero 4-bets to $2, 1 fold, BTN calls $1.30

Flop: ($4.30) 8♦ 8♣ A♣ (2 players)
Hero checks, BTN bets $2.15, Hero calls $2.15

Turn: ($8.60) 6♥ (2 players)
Hero checks, BTN bets $4.30, BB (Hero) folds