France is really dog friendly (small dogs at least)
If coming from outside of Europe, you will need to have your dogs medically checked and approved (and probably chipped) and have his own sanitary passport.
IMO, I would leave the dog at home; less stress for you, less stress for him
Dogs are not allowed in museums and other public service buildings (post office...)
(addition) Dogs are not allowed in grocery stores and other places selling food (for sanitary reasons), but allowed in outside markets.
Dogs seems to be allowed in restaurants, but check before (if there is a terrasse, then you should be ok if sitting outside).
Small dogs are allowed in public transport (bus, metro, RER) if they are in a dog carrier.
For hotels, you need to check with each of them.
Dogs are allowed in parks and squares (on the path ways), and must be held by a leash.
You are responsible of picking out the trash.
also see this (for paris, but should be fairly the same for other regions)