While it does use "bite", which would seem to be a small amount, it usually means to go out (to a restaurant) and get something to eat. It does not refer to specific food, and I usually hear it with "to eat" (as in "grab a bite to eat"). Your first example doesn't really sound natural since it's usually used to mean "go out to eat" and the third one is definitely wrong - the preposition "to" doesn't work there. The second example is fine, although I would expand it to "I'm so hungry; I'd like to grab a bite to eat." You generally hear this being used in a question, e.g. "Want to grab a bite to eat?"