What you have suggested is possible. Murakami's sentence is an example of right-node raising explained here:
Use of first が in 町の後ろに山が、前に海があって、
But usually you should not do this in a simple sentence like 「料理が、掃除ができる」. This sounds unnecessarily literary or even theatrical to me. This might be fine in lyrics or as a catchphrase of a robot, but in most cases ...
What you might want is 先生、[何か]ある？.
の is used to confirm what you observed, not for simple questions.
何かはある？ is an expression to point out that the other person said s/he doesn't have something but not necessarily anything.
edit; [何か]はある？ 1. asks if it's normally available, if not immediately, 2. confirms it for the listener to make sure to have it. 3. shifts ...
I can't explain it correctly in grammar terms, but here is simple explanation why you can't use 本を読んでいる人が誰ですか？
Case 1. You see man reading book and want to know who is it (is it Takada-san?), so you asking 本を読んでる人は高田さんですか. And answer will be はい、高田さんです。 Yes, he is mister Takada.
Case 2. You see two men. One of them reading book, another one drinking juice. ...