I'm afraid I can't answer definitively who manned the French galleys, other than to say that they were not slave galleys -- the French used several manning systems over the years for their navy, often conscripting men into service. But I can say that the Mediterranean galley fleets were instruments of war, though the French galley fleets did not often directly fight other galleys -- the Spanish and French galley fleets never engaged in battle at all, for example.

Galleys lasted in the Med longer than they did in more northern waters both because galleys are fairly useful ships, and also because the Mediterranean is (comparatively) calmer than the North Atlantic. Although galleys could be used in northern waters -- the French galleys participated in an expedition against Antwerp and raided Harwich in the early 1700s, during the War of the Spanish Succession -- they were highly useful in the Mediterranean especially in amphibious expeditions or in actions close to shore. Galleys are of course able to move independently of the wind, and are shallow draft ships, so they can get to places that sailing ships may not be able to given wind and weather conditions. Galleys of this period were generally armed with one large gun right in the bows of the ship, flanked by lighter guns, and could carry decently heavy crews (say 250 rowers and a mixed crew of 250 other men, in some combination of sailors and soldiers) suitable for a variety of missions.

That said, the French Mediterranean galley fleet probably lasted as long as it did (it was disbanded in 1748) partly because it was a prestige fleet, having been massively expanded by Louis XIV during his reign.