The reason you don't get a lot on prop vs prop fighter combat is that there wasn't much of it. There were some F-51 vs. Yak-9 fights, as well as F-82 and F4U engagements with Yaks or Il-10s as well. But when the F9Fs and F-80s showed up, the ballgame was over.
There were a few stray fights later on - MiGs vs. F-51s usually went the way of the MiG, but there are some incidents in which F4Us and Sea Furies at least got a draw out of them.
Big reason why seems to be once the KPAFAC was shredded Kim Il Sung appears to have wanted a force in being rather than a force in the field. They appeared on occasions but due to the inequality of any inexperienced KPA pilot vs. a trained UN one he preferred to avoid a fight. Hence the use of propaganda about "Hero Kim" and "Hero Ong" and the female MiG-15 pilot who singlehandedly shot down all those Sabres over their base. (You can win a lot more in fictional reports than real world combat!)
The Russians have been publishing histories of a lot of units and aircraft but must say I personally haven't seen much on P-63s. Pokryshkin, the 2nd ranking ace, did score the majority of his 59 kills in a P-39.
The choice of the 37mm was for killing bombers more than anything else. It was a slow-firing (250-400 rpm) gun with a low (690 mps) muzzle velocity, and one thing which doesn't come out too often in the reports is the fact that most of the time the better Soviet pilots used either the twin 23mm or the single 37mm gun in fights. That both gave them twice the "staying power" (about 12-14 seconds of ammunition) as well as the ability to engage fighters with 23mm and bombers with the 37mm.