"Are you sick?"
Beteg means to have an illness, indeed. Talking about rosszul is more feeling unwell.
Is the "ssz" pronounced like "sz"? I don't hear the "s" sound in the recording.
'Ssz' is a double-sz, so it's pronounced like a long 'sz'. If you double a digraph (or a trigraph), you only double the first symbol: cs -> ccs; gy -> ggy; zs -> zzs; dzs -> ddzs; and so on.
You just have to watch out with compound words, like karosszék - armchair, which consists of the parts karos - armed.. erm, "with arms", and szék, chair. There is actually 's' clashing with 'sz'. But since those sounds are difficult to pronounce so closely together, it's mostly spoken as an actual 'ssz' anyway.
What is the difference between Vagy and Vagyok here? I don't seem to be really able to choose which one to use
It's conjugation! Hungarian is full of that:
- vagyok - I am
- vagy - you are
- vagyunk - we are
- vagytok - you (pl.) are
(The verb "to be" is a bit strange in Hungarian, since "he/she/it is" and "they are" usually aren't translated, but you rather say something around the lines of "A ház zöld" - "The house (is) green".)
Is gy somewhere in between g and j (English pronunciation) in terms of how the tongue is positioned?
It's at least very odd. What you want to do with this question is finding out someone's condition: sick, or not sick? So rosszul is the focus point here. And in Hungarian the focus appears in front of the conjugated verb: "Rosszul vagy?"