can you say sakitkah kamu?
You can. But it is rarely used.
The most common informal phrase is "Kamu sakit ya?"
Why is the question without apakah?
You can use "Apakah" in this case ... both have same meaning
why do you not need berapa in this instance?
Berapa is "how many" or "how much", it has nothing to do with a yes or no question, the conjunction you may be thinking is "apakah", which can, but is not required to, be used here.
to indicate a question is being asked - I think apakah or apa is needed in this example