Since you won't be in France, you won't be working in France. You therefore do not need permission to work in France.
Furthermore, as a contractor, the business relationship is between your corporation and the French company. That your corporation has a sole employee does not change the fact that you are employed by that ...
A posted worker would be someone your company sends to France to complete the work. In that case, for a 6-month contract, you would need a residence permit or long-stay visa (which implicitly gives you the permission to work) and I think that's what you are being asked about. As long as you are physically in the US, you don't need that.
If you need to come ...
Even though many (possibly most) women take their husband's name upon marriage:
This is far from universal, and definitely not mandatory
On all official paperwork (ID cards, passports), they keep their original name, with the addition "Epouse X", rather than actually change their name.
So no, there's absolutely no requirement for you to change ...