[kghose]

1. You can apply for permanent residency under the extraordinary individual or highly qualified individual (if you have a specialty profession e.g. STEM)

2. Academic and non-profit jobs don't have a cap. You could take such a position and then apply for permanent residency.

3. You can apply for the green card lottery (I'm assuming you are born in France).

[Synroc]

I like 2. and 3. Could I do 3. while doing 2?

[kghose]

Yes, as far as I know the lottery is independent of anything else, but you should check the USCIS website and consult an immigration lawyer.

Be careful with the lawyer: try and get one through a recommendation of someone you trust who has used their services. Because of the high demand for immigration into the United States, there are many unscrupulous people involved in such matters.