Feasibility of moving from US to France for a postdoc opportunity
I recently did exactly what you were describing. Spent two years in Paris as a postdoc.

1) Not speaking French will be an issue because of collaboration. Some will be very open to working with you, but many others will not. Do not be surprised when group meetings are conducted in French. The more international the lab, the better the experience will likely be. Most of my collaborators and friends ended up coming from other countries outside of France; though they all also spoke French.

2) In a major city like Paris, day to day living is not really an issue. Getting around the city and running basic errands are really not that difficult.

3) Getting the work visa is a non-issue. The host organization will handle this and provide you with the paperwork (though probably not promptly). At some point you will have to visit a French Embassy in the US to file the paperwork in person. You will likely get something called a Scientific Visa which is very permissive in what it allows. 

4) Here is the bad news. Bureaucracy is an absolute nightmare in France. If you want to navigate it, you will either have to speak very good French or have a friend willing to help you. This help will require multiple full day commitments, so it is a lot to ask. Once you arrive, you are supposed to apply for a resident permit and are technically not allowed to leave the country until it arrives. Mine took over a year and I know people who have gone 2+ years without receiving one. In the meantime you get a paper stating you are waiting on this permit. It expires every 3 months, so you have to present yourself in person to renew it (actually a renewal usually takes two visits). I never received proof of health insurance. That means I could receive healthcare, but I had to pay full price. From the people I know, this is a very common situation. They will still deduct money from your paycheck every month though.

5) Obtaining an apartment and bank account in France are quite difficult too. It is sort of a Catch-22 situation. Before arriving, I was put in touch with Science Accueil. They help setup foreign scientists with apartments, bank accounts, and things like that. Hopefully you will also have access to something like that.

Living in France as a non-EU citizen is very difficult. I don't regret doing it, but I don't think I would do it again. There are positives though. Your situation may be different, but I received 9 weeks of vacation per year (including holidays). It is easy to travel throughout Europe and that experience is amazing.

Good Luck!