What is the process for PhD applications and contacting professors in France? I would like to apply to a PhD program in France, but I am having trouble understanding the admission process.  How should/could I submit my application? 
Is it proper to contact the Professor I would like to work with and ask about any openings? If yes, what should I include in my mail?

I am only familiar with the admission process in US institutions.
TL;DR: in France you have to find first the supervisor, then you apply for a funding. If you have a funding of your own, you have to find a supervisor to be admitted. In both cases : you need the supervisor first.

There is various things to know when you apply for a PhD in France :


It is forbidden to be a PhD without having funding. In France a PhD student is considered both as a student and an employee.
The funding can be a state funding, an industrial funding, a funding on a research contract (either from a national funding agency or a company), or you can also do your PhD while working elsewhere (some teachers in secondary schools are doing their PhD this way).


Industrial funding and research contract are basically given to the supervisor, who can choose alone amongst all candidates. So for PhD with those fundings you have to contact directly the targeted supervisors.

State funding is given by a committee to a bundle (candidate, subject), this means that a professor, with a subject, has to find a candidate and then propose to the committe this candidate on his subject. Then the candidate is on his own : (s)he will generally have to send an application letter, with a reference letter from the supervisor, and if (s)he is shortlisted, (s)he will have to make a short presentation (either on location or using skype or a similar service).

Last case: you are funded on your own (job, external funding for foreigners). Even in this case, the procedure is that you find the supervisor, then you apply.