All public school teachers in France are fonctionnaires (civil servants), and you are not allowed to become a fonctionnaire unless you are French. (Believe me, as an aspiring English teacher, I have done my research on this). You are technically allowed to take the civil service test (the CAPES), but you need special permission from the départment in which you wish to teach, and given that you can't even be hired afterwards, the point is basically moot. You can take the CAFEP, which is essentially the same test but for private schools, but in reading the literature associated with it, it seems to be heavily oriented towards people of the religious persuasion who want to teach in Catholic schools, which makes me hesitant as an atheist Jew. Your mileage may vary.
To teach in a university, you have to be an enseignant.e-chercheur.e (technically in France, this happens automatically if you receive a French doctorate and are hired by a university in an official capacity, but I have no idea what the steps would be for someone with a foreign degree) or agrégé.e (which means you have passed the agrégation, another civil service test with the same pitfalls as the CAPES mentioned above). This is something you could possibly pursue, but I would assume you'd have to get in touch with the relevant personnel at a specific university (or universities), for as a second-year Master's underling, I unfortunately wouldn't be able to say specifically how you could make that happen.
From what I've read on this forum, any sort of medical degree is very hard to make work in France, as qualifications are very strict and, I assume, very distinct from those in US. You also have to have impeccable French to be able to work in the health care industry in France, especially nursing, where you're in direct communication with the patient.
Sorry to be a bearer of bad news.
