vocabulary
I need to ask you a question. = (less commonly) I need to ask a question of you.
(Merriam-Webster’s Learner’s, ask #1)You cannot say ‘ask to somebody’: *I asked to my friend what had happened.
(OALD, ask #1)
When you ask questions, ‘ask to somebody’ cannot be used, says OALD. What about the next example? It’s the pattern of Merriam-Webster’s, but whose bracket ‘less commonly’ makes me to wonder if this sentence can be used.
[next example] I asked what had happened of my friend.
As commented by @Walter, in OP's context, fancy is a synonym of imagine, believe. But the usage is becoming increasingly "dated"...
Many of the above instances will be the idiomatic exclamation "Well, fancy that"! (equivalent to "Imagine that!", "What a surprise!", "How strange!", etc.). In OP's context, Dumbledore could quite naturally have used any of those (possibly preceded or followed by something like "I didn't expect to see you here!").
Note that Dumbledore is an elderly British schoolmaster in a relatively "formal" situation (he's greeting Professor McGonagall, one of his staff who apparently happens to have adopted the form of a cat).
Idiomatic expressions as used by Dumbledore won't necessarily be so natural for younger speakers, and per @StoneyB's comment below, even Dumbledore is speaking ironically here. Brits in particular still use fancy = like, want, be attracted to quite naturally today, but fancy = imagine, believe, think is now largely confined to ironic/sarcastic contexts.
She was obliged to make the interview shorter than expected.
had to is the idiom have to cast in the past tense. Have to has the same meaning as must = “be obliged to, be required to”, and has slowly been replacing must over the past two hundred years. It is a much more flexible construction because unlike must, which has only one form, have has a full complement of finite non-finite forms: present, past, infinitive, present participle and past participle.
cut short is a literal expression employed figuratively: she ‘cut’ the interview off before the appointed time, so it was shorter than planned.
Best Answer
If you're going to use the "of" form, you would need to reference the person being asked:
I asked of him what had become of my friend.
But this would be something you might have heard in Shakespeare's day.