(이)라고: Used when you intend to convey a message to an or a group of audience.
So for the verbs 말하다, 쓰다, 생각하다, 믿다, 부르다, etc.
~라고 is often used.
오빠가 이 걸 안 할 거라고 그랬어/말했어 ==> I said my brother won't do that
병이 나아질 거라고 굳게 믿어 ==> I strongly believe that the illness will get better.
1. Why is (이)라고 used here as in "누구라고 해요"?
In short, it is a habit. As it is not you who are speaking, it is a good habit to use quoting.
A possible reason of having such a habit is to ensure the accuracy of your speech. The one on phone claims to be a certain person, however his identity is not confirmed yet. In such a case, communication gaps may happen if quotes are not used.
2. Can I just use object particle in replacement of (이)라고?
Yes, grammatically it makes sense to do so, and Koreans will understand what you mean. BUT personally, I've never seen native Koreans who don't use quoted sentence for the abovementioned case.
In your case, as the verb is 쓰다(write), you should use (이)라고 to make the sentence natural. "여기에 뭐 써야 돼요?" does make sense, but only when 쓰다 is used to mean "to wear (hat)" or "to use".