Is it possible to use US-made engines in Russian made air-frames?
Re-engine an existing airframe is such a complex undertaking, in a way that so much R&amp;D is required that the entire aircraft ends up being reverse engineered to make it work. Or on some occasions, the airframe is first licensed or reverse engineered, after which, experience and expertise is gained to enable the re-engine effort.

China is notable in this area due their Russian lineage of airframes and a long lasting desire for western engines, and a mixed relationship with both. India is in a similar position but I'm not aware of any such attempts. 

Notable examples:


Y-8-F600(Antonov An-12 clone) with PW engine, failed
Sabre II(Mig-21 derivative) with GE or PW engines), failed. 
Shenyang J-11(license production Su-27 derivative) with Shenyang WS-10 engine (reverse engineered CFM56 derivative), successful.