The Navy shot down at least three aircraft the Soviets indicated were "non-players" and the Air Force had one.
Several F4Us shot down an A-20G from the 136th Torpedo Bomber Regiment on 4 September 1950. The Soviets filed a complaint and the US returned the bodies.
On 18 November 1952 the Navy CAP engaged four MiGs from the 83rd IAK and shot down two, damaging one other. The Soviets went ballistic and accused the Navy of waiting for the two pilots downed to freeze to death before "rescuing" them.
On 27 July 1953 CPT Ralph Parr spotted an Il-12 transport from the 83rd IAK that was transferring pilots and shot it down with a full loss of life (23 KIA, I think). The Soviets were furious about this one as it was clearly marked as Soviet (e.g. Red Stars) and in point of fact they sued Parr in the World Court for wrongful deaths and liability for the loss. It took several years but the Court finally ruled the aircraft was in what cound conceivably be considered a war zone and ergo took its chances.
There were some freelancing expeditions like the adventures in shooting up Soviet airfields like the one where the P-63s were stored, but most were taboo. The main fight that came up later was when the pilots crossed the Yalu to engage the MiGs in their landing runs and eventually lost some Sabres. At least one ace pilot was charged with breaking orders but when the facts came out about how the orders were flagrantly ignored the USAF eventually let it drop.