I would 3bet/fold preflop to $2,20. I don't want to play a hand that can easily be dominated oop without initiative. 3bet or fold.
With your preflop line, postflop, you have the nuts so the question is "how do you get the most value?"
C/C F, C/R turn seems ok. But I would probably go with C/R Flop, because I want to build up the pot and villain would call with two pairs most of the times, AQ, AT, KQ,KT also. AA, JJ, KK will raise, so you have an easy all in till River against sets.
So C/R F to $3. Then value town. If he raises Turn, push.
About your Turn line, I understand you want to get the most value out of your hand, but c/rai it's a clear sing of the nuts. Better c/r big and put it all in on the River.