Which sentence is correct?
- Who don't want to lend his umbrella?
- What Paul doesn't want to lend?
- What doesn't Paul want to lend?
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The default value of "who" is singular. So if you don't know what "who" refers to, you should treat is as singular. So your first sentence is not idiomatic and should be:
Who doesn't want to lend his umbrella?
There's an exception to this rule, of course. When it is presupposed that the answer is plural, the plural agreement can be used, eg:
Who haven't yet handed in their assignments?
Your second sentence is not grammatical because it lacks a subject-auxiliary inversion. It should be "What doesn't Paul want to lend?", which is your third sentence.
However, "What Paul doesn't want to lend" can be construed as a fused-relative clause (also known as a nominal relative clause), similar to "that which Paul doesn't want to lend". A fused relative cannot be a stand-alone clause. It should be embedded in another clause. The sentence below, for example, is grammatical:
What Paul doesn't want to lend is his umbrella
It can also be a subordinate interrogative (embedded question):
I don't know what Paul doesn't want to lend
The meaning of the sentence is similar to that of I don't know the answer to the question "what doesn't Paul want to lend". In a subordinate interrogative, the question is embedded in another sentence, in which case, the subject-auxiliary inversion is excluded.
The first sentence should be:
Who wouldn't want to lend his umbrella?
The second one isn't a question. Its more like a statement about something that Paul doesn't want to lend.
What Paul doesn't want to lend.
And the third, is a rhetorical question. Its a question to which an answer is not expected. It means that Paul would lend everything that he owns.
What doesn't Paul want to lend?