As a Parisian, I can tell you that you can shoot inside the Opéra Garnier. However, you may have to buy a guided tour to be allowed to shoot. I don't think tripod is allowed but you sure can use a monopod or any other device which is not too big.

Montmartre, as most touristic places, knows its pickpockets, thieves, riff-raff... Close your bags and hold them tight, and do the same wherever you go in Paris (métro —subway—, buses, etc.). The Eiffel Tower is crowded with men who will try to force you to buy their crap at very expensive prices. Make sure not to be surrounded by little groups of children : while one asks you the time or a direction, the others will pick into your pockets and rob you.

Besides that, if you're not alone, you got nothing to fear, late or early. Keep your gear hidden as long as possible, not to tempt thieves.

Hope I didn't afraid you, enjoy your stay. Parisian shop/bar owners and taxis are sometimes rude but street people are welcoming. As AJ said, I recommend the Musée du Louvre and le Jardin des Tuileries which leads to the Louvre. Make sure to visit the Musée d'Orsay which is gorgeous too.