You could just cheat and say that animals are different in her world, or that it's a magic squirrel.(just look at Hiro Mashima's "dog")
Could you rephrase that? The part about classes, I mean. As for story, the traditional start to a D&D campaign is "You are complete strangers all sitting in the same tavern, when a mysterious person stands up and says they want to hire some adventurers.", so...
This depends on how specific the premise is, and how much it restricts the kinds of characters that would fit.
Most GMs, particularly for D&D, just say "play anything". Part of this is because D&D classes, feats, etc. have some fluff built-in, which stays the same in any game of D&D you play. Plus character-building in D&D feels more like a puzzle, so a player can feel proud of something they put together well, and want to show it off. In a game like M&M you have to do most of the description yourself, which makes it harder to get attached to a build.
Then you have games like FATE (including Dresden Files RPG), where your character description is the biggest part of your abilities, and part of character-building is writing your history with other members of the group.
I've heard good things about Way of the Wicked.