It’s definitely a casual construction. It seems i typically think of it in the form “gimme it” (which is mostly associated with children), so it goes with the *grabby hands* thing.
More data! Where are you from, and would you say it is mostly children’s language, or just something belonging to a casual register?
I’m from the mideastern USA (northern Virginia, specifically). And… hmm. I think it’s mostly children’s language? Or language used by adults who are emulating a childish degree of enthusiasm/greediness/whatever? Like, i’ll hear adults say it, but it’d be in a silly way. (As in, telling you to gimme your dictionary from across a giant ocean. XD)
Gimme that, on the other hand, is just casual register, although usually accompanied with “would you?” or something to make it more polite and less demanding.
Moral: Language is weird.
Kimchi and Justice 