Learn English – “Will have already” vs “Will already have” – which is correct?

It sounds horribly odd to my ears (probably worth noting the group in that song is not native-English speaking). Crave is (virtually) always a transitive verb in Modern English and needs to have an object. The reason that the noun form works with for is that nouns will need prepositions to add in objects like that. For example, The company prints Ø the books versus There's a new printing of the books.

As Peter Shore points out, the OED recognizes its use, but its usage is quite rare these days. Searching around with some other phrases besides I crave for (since that's taken up by song lyrics), almost every instance of crave for sounded off to my modern ears except when for was the head of a clause or when the object craved is intended for someone else:

• Someone craves for someone else something.
  • in other words: Someone craves something for someone else.
• I crave for someone to do something.
  • but not *I crave someone to do something.

For examples of these you've got Rest he craves for you in His bosom ¹ - …he craves for you to f— off ²

So you can use it as it's certainly been used that way, but recognize it will come off as antiquated or weird to most people.

Normally questions with "Did you ...?" refer to a definite time in the past indicated with adverbs or adverbials such as yesterday, lately, last week, last year etc.

And normally questions with "Have you + past participle" ask about a fact with no regard as to when it was. A professor of literature might ask a student "Have you read Hamlet/Beowulf? The professor wants to know whether the answer is yes or no.

These are the normal uses, but in real life questions asking for facts are also done with "Did you...?" as has already been said in the posts above. Why? Well, I think in colloquial language it's a bit cumbersome to analye such questions and in fast talk there is often no time for such analyzing. So the grammar system is often simplified and speakers use "Did you...?" when actually "Have you ...?" should be used.