Why is there a что needed in this sentence?
It is a useful pattern for "..or what?"-type sentences where you are not quite sure of your own conclusions. Thanks for noticing it, I was going to delete all of them from the course for a long time.
It is to emphasize the surprise. The same as English "what" in the beginning of the sentence.
Okay thanks, but it seems strange to my non Russian ears. I mean, do you have to say half a sentence and then get surprised? Also, would this sentence be correct without the что?
No, you say the whole sentence as if in disbelief.
Without "что", this becomes a more neutral question: Doesn't the brother have a table at home?