It seems pretty clear that yes. Probably won't do the warning shot until a EU city where a french embassy/school is present is bombed, but clearly yes.
Post-95 french doctrine refuse the use of nuke as a tactical weapon, so i guess supply lines or military airports would be the targets of subsequent nukes, but we have generals who grew with the idea of using tactical nukes, and i'm pretty sure the possibility is still studied at St-Cyr.
French will say "we won't bomb Russia, but do you think we can bomb your territory, close to the border, on this train station/close to this airport?" It will probably be refused, until its _really_ needed.
Russia (and US) doctrine of only using nuke if their territory is nuked does not work in an offensive war.
You're correct. They wouldn't. The US and France would not nuke Russia if it invaded Poland or Germany. But that isn't necessarily true if Russia invaded the US or France (and had a chance of winning).
A similar dilemma: will the US risk New York to save Paris?
France did not believe that the US would risk New York to save Paris.
So France developed it's own nuclear deterrent.
I leave the rest as an exercise to the reader.