In "proper" conversational and written English, the first three should all be "quickly" (though some dialects might use "quick"). In #3 more people of all persuasions might say "quick" in conversation, or even in informal writing (though never in formal writing). I might myself.
Given the way you've punctuated #4, this, again should be "quickly" in "proper" English in my opinion. "Quickly" qualifies "call" adverbially in the usual way, but there's a slight pause before the "quickly" sentiment is expressed (kind of like "You need to call her ... and quickly.") It is possible for "quick" to be an exclamation that urges the person to hurry up, rather than an adverb qualifying "call". I think that's what you're getting at, but to me an exclamation mark is required, and I'd probably write it as two sentences: "You need to call her. Quick!".