My Le Creuset Dutch oven is best used on my wood stove. The steady heat, spread evenly over the bottom of the pot makes for a fantastic braise. In fact, I've just taken some short ribs out of the freezer to cook on the cookstove tomorrow. Chilly here in Ontario too.
Yes, of course. Unless your oven's spot goes >500F or you leave it on--dry--for days.
Most modern ECI is not enameled on the bottoms. But if yours is, theoretically you could get some minor crazing of the bottom. Almost always a cosmetic issue not affecting longevity.