"It's me/I they want!"
You may be confused because of the existence of sentences like the following:
"I am [whom they want]."
Here, the subject is "I." The complement is the relative phrase "whom they want." The relative pronoun "whom" is prescribed in this context because it is the object of the relative clause.
The sentence you're asking about has a different structure:
"It's me/I [they want]!"
This sentence is definitely not equivalent in structure to "They want me," so just because "me" is the object of that sentence doesn't mean it's an object in your sentence. I actually don't remember if "it" or "me/I" is considered the subject in sentences like this (I'd guess "it" because the verb agrees with it), but it doesn't make a difference for your question because the main verb is "to be". Prescriptively, the complement of the verb "to be" is supposed to have the same case as the subject, which in this case would be the nominative.
Unlike "whom," "I" and "me" are personal pronouns, not relative pronouns, so it's not possible for "me/I they want" to constitute a clause.
Since "me/I is not part of the embedded clause, the case of the pronoun is just determined by its role in the matrix clause (similar to your example "It is I whom they want" or the situations discussed in the following questions: Which is grammatically correct: "Let he who..." or "Let him who...", Is it acceptable to start an emphatic sentence with "It is he who…"?).
So the prescribed form would be “It's I they want!”
But "It's me they want" sounds grammatical to me, and I'm pretty sure to many other English speakers, so that's also OK (see Which one is correct to say: "It's me" or "It's I"?).
Interestingly, though, "It's I" is still more common than "It's me" when followed by a relative clause starting with the subject pronoun "who," according to Barrie England's answer to the following question: It is I who am at fault? Possibly, the reason is because many people find sentences with first-person verb agreement but not "I" (such as "Remember me, who am your friend”) jarring.
“Who is it you asked?”
This is just the question corresponding to the statement "It is __ you asked." When you form a question, the prescriptive rule is that the interrogative pronoun takes the same case as the corresponding word in the equivalent statement. (In normal speech in real life, the situation is more complicated since we can say "It's me" but not "Whom is it?")
Since we established that the pronoun in "It is me/I you asked" is not a grammatical object, we would use "who" and not "whom."