Suppose, someone buys 300 Pratt & Whitney engines from the USA and 150 Mig-35 air frames from Russia. Then, they use their in-house expertise to fit those engines to those air-frames.
What challenges would have to be faced by the user?
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Re-engine an existing airframe is such a complex undertaking, in a way that so much R&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.
We're not on law.stackexchange.com, but from what I can tell, the legal consequences would be limited to losing some benefits from warranty/support agreements.
There is no reason to do this. The engines on modern Russian fighters are quite good and among their selling points. You'd be concerned with upgrading their avionics - which are, of course, one of the most tightly-controlled export items - not their already excellent flying qualities.
This is not possible. For airliners or other pylon-engine configuration aircraft, re-engining is easy. For fighters with internal engines, it's not. The airframe is built around the engines, with special ring frames that allow room for the engine while carrying through the wing loads.
The engines are not geometrically compatible. None of P&W turbofans that could suffice to power a MiG-35 would fit in it.
TL;DR version: To paraphrase the old joke, you want a fighter that flies like a Flanker, sneaks like a Raptor, shoots like a Rafale, and maintains like a Gripen... not the other way around.
The MiG-35 is Russia's "low" component, similar to the F-16 in the US arsenal - lower cost, lower maintenance, high but not best achievable performance, and in its case also good improvised airfield capability. As part of that, it's a mid-size airframe, with smaller than average engines, optimized for efficiency and durability. The only US-built engine that could both fit and power the MiG-35 is the GE-414, after some modifications.
However, it's not the best possible engine for the MiG-35. The latest version of its own engine, the VK-10M, offers more thrust than the GE-414, has vectored thrust versions, and is a perfect fit for the frame. If you want the best, you'd want a custom engine, like an enlarged higher-bypass version of the EJ-200 (that's doable).
But, then, if you want the very best, you'd probably start with the Su-35 or the F-35. The MiG-35 is a workhorse.
If the question is not limited to specific airframes, the Irkut MC21 airliner is planned to have two engine options : Pratt & Whitney PW1400G (American) and Aviadvigatel PD-14 (Russian).
So the answer is Yes, if you include airliners; you can even order them from the factory that way. First deliveries were scheduled for this year : I suppose that may be subject to change.
(there were also plans to give the Sukhoi Superjet a P&W engine option; these seem to have fallen by the wayside at the moment)