Oh, yes. they are slang expressions, certainly - but "having a bird," "having a cow," "flipping out," "blowing [one's] top" all basically mean the same thing in my part of the world. I wouldn't define them as "is crazy" in the literal sense of "Is mentally unstable and should be institutionalized," however, but rather as "losing control" or "becoming extremely enraged/visibly frustrated" or something like that. [Native US English speaker here.]

@Soglio: It isn't necessarily a permanent state. It's something you say when people do things you consider somewhat crazy. Anger or an emotional eruption is normally not implied. "Eine Schraube locker haben" and "Einen Vogel haben" are synonymes in German and seem to fit well with "to have a screw loose". (At least that's what I get from Pons. The existence of this phrase in English is something I just learned, so I dont' have much experience with using it, yet. But I'll try it out on a bunch of people and report back :) )

So, not clinically, certifiably insane, necessarily, just nutty. OK, got it. I'll look forward to the results of your empirical research (but do be careful ;-).