They know all those things to the same extent to which they know about them in the USA.
The FAA system is one where trust is vested in suitably qualified individuals e.g. an A&P/IA or a Part 145 Repair Station to do the release to service.
They have very active offices abroad who inspect the FAA scene out there. Much more pro-active than the local CAAs especially the UK one.
Trust ownership is probably impossible to stop - even if they wanted to. At a very basic level, they cannot prevent me selling my N-reg plane to a US citizen (living anywhere), for any amount (say $1), and he loans it back to me. Is the FAA going to ban lending your plane to somebody?
I don't know who you are trying to scare, GEP.
I see some seriously dark clouds on the horizon. For example that bogus project called PRNAV, which has the potential of wiping out a huge chunk of the IFR community bigger than anything EASA might have done, due to the sheer cost of ripping out and replacing loads of perfectly good avionics, for no technical reason whatsoever. Just to get the "right paperwork".
But I am not worried about the FAA screwing the N-reg community because it is all over the world, and is a useful component of US foreign influence which costs the USA peanuts.