You are correct that you can use "What about you?" (also "How about you?") to repeat a question to a second person, in almost any case where it's about a person's situation. That is, it replaces the person who is the subject of the first question with the second person you're speaking to. Literally, "What about you?" means, "Does that situation (or condition) also apply to you?" For example:
Bob, would you like some more wine? Mary, what about you?
This asks Mary whether she would also like some more wine. It effectively replaces the subject of the first sentence: "you [Bob]" with a new one: "you [Mary]."
Mary, are you working next week? How about you, Bob?
This asks Bob whether he will be working next week.
You can also ask someone "What about you?" following a statement (about yourself or another person) to ask whether that statement also applies to them. Again, the new person replaces the subject of the first sentence.
I'm really tired after last night's party. What about you? (are you also tired from last night's party?)
Mary loves cats. What about you? (do you also love cats?)
You can't really use "What about you?" for "a question, any question," because the question has to refer to a person's situation and have a person as the subject. For example:
What time does the game start? What about you? (wrong)
This doesn't make sense because you're not asking any question about a person's situation in the first question and the subject is "what time," not a person.
If you use the phrase in the wrong place, it can sound ridiculous:
What should we have for dinner? What about you? (wrong or terrifying!)
The only way this sentence would be correct is if you are suggesting that you literally want to eat the other person for dinner. Here the subject of the first sentence is "What [is for dinner]," and you're replacing that subject with "you [are for dinner]."