That's not because the french police is especially militarized or anything, it's just that they are under threat. Swedish police patrolled the streets and malls with submachine guns this christmas too, it happens more than you know. Islamic terrorism is a real and ongoing thing.

This is for show only. In very few occasions are they allowed to use them. Ammunition are actually stored safely away. At least that was the case few years ago. The official report about the Bataclan attacks mention such unit not being given clearance to get amo.

Sounds stupid in that case, but I believe they did the maths properly when balancing the chance of armed illiterate young men in public killing innocents versus saving the day.