Shouldn't there be a contraction? Vă rog, nu m-întrebați. Plus, are the contractions obligatory?
Contractions are not mandatory, with very few exceptions.
The way you contracted the sentence is false. You can contract it like this: Vă rog, nu mă-ntrebați!
Which BTW in vocative/imperative (should I say also ”caesar-ative”? :P hehe) is ”Întrebați-mă” (”Ask me!”) - the only possible form, and obligatory contracted.
Contractions are most commonly used when speaking in the past tense form: Nu m-a intrebat(he did not ask me).
So here mă becomes m- because of the verb avea or because it starts with the "same" vowel (ă - a)?
It becomes m- (as far as I can tell from what I hear) because it is the same vowel, it is hard to say "ma a," so it is shortened into m-a. It is the same with many past tense forms. This probably came about through a difference in speaking(people shortening by skipping a vowel) that was made into a written rule.
That is indeed true, very good assumption. We use contractions in a lot of places where pronunciation of double vowels could be hard, in fast speaking, but the contractions are never mandatory, with very few exceptions ("într-o"), and not only related to verbs or past tense. Compare: ”Mi-ai lovit-o-n cap cu-n ciocan” (”you hit my (feminine gender object known from a former sentence) in the head with a hammer”), here also the ”cu un ciocan” (”with a hammer”) contracts to elliminate the double oo-oo sound.