Almost - I would say long 's', not double.
Once we are here, there are compound words (with some determinedness and luck you will create them yourself soon) , in which the first part ends with 'sz' and the second part begins with 'sz' - we write the two 'sz' in full, no abbreviation. E.g. 'szászszármazék" (= of Saxon origin), etc.
You know that it means you because of the verb form: vagy only appears with a second person singular subject, like you or, old-fashioned thou.
As for "Is he/she sick?" You would use
In both Rosszul vagy? and Rosszul van? you can, but don't have to add the subject:
Te rosszul vagy? Ő/Ön rosszul van?
Note that you cannot leave out the verb with the third person suffix here.
Correct, can't leave it out at all when we are asking about somebody'swell being. Of course there may be exceptions when the context is given previously.
"How are you ? (Are you) sick ?" could be translated as "Hogy van ? Rosszul ?"
But "Rosszul?" on its own is lacking subject / pointer. Like if I made funny face and you would ask me "Rosszul?" I may ask back if you mean something went bad or if I am feeling sick, etc.
Third person van is not implied in your last two examples. You can't drop the verb in the third person when you're using an adverb like rosszul. When you're using a predicate adjective like rossz you can:
Ön rossz? Ő rossz?
But not with adverbs or phrases describing the space or the time that something is happening at:
Hol van? = Where is it? Mikor van? = When is it?
The reason is that when you ask a question, the thing you question has to precede the verb (linguists call this focus).
When asking Are you sick?, you ask about being sick or not, so it has to come before the verb in Hungarian:
It's similar with questions words like ki ‘who’ and mi ‘what’:
Ki vagy? ‘Who are you?’
Yes, unfortunately it would sound weird, even to the extent of impaired comprehension. It is not grammatical, either. So, in this case: adjective/adverb first, verb second:
Jól vagy? - Are you okay? Rosszul vagy? Are you not feeling well? Elégedett vagy? - Are you satisfied? Gazdag vagy? - Are you rich? Éhes vagy? - Are you hungry?
That isn't necessary if the meaning is still there. Making spelling/grammatical errors that doesn't affect the meaning, should be accepted as we're learning Hungarian and not English. Making an is/are error in the Hungarian course should be accepted, but making an is/are error in the English course should not be accepted.