@Jeff: Yes! The name didn't ring a bell (maybe it had a different name in French? Or maybe I just didn't remember it) but this video is DEFINITELY it. I remember the boards, cards and so on... certainly don't remember any of the monsters being red or anything, but for all I know my brothers painted them all, I remember they were into painting miniatures.

EDIT: I've just done a bit more checking to find the French name. Well, Dragon Strike was never released in France, however Heroquest, which is apparently a simpler version, was. That's definitely the game. As soon as I saw the name, I thought "YES! YES!", and pictures I saw corroborate that.

Thanks so much for helping me figure out what that game was.