gesund is really health related. Actually the sentence sounds a bit odd to me. I would use it like Bist du wieder gesund? (Are you healthy again), because the state of being healthy is considered the usual case. Otherwise I would ask Bist du krank? (Are you sick/ill). gesund IMHO only applies to "real" illnesses like flue, cold, infections, etc. If you break a leg, if would feel weird to use it.
If someone has a general problem be it emotional or the mentioned broken leg you would ask something similar like "Are you okay", e.g.
Bist du ok?
Geht's dir gut?
Alles in Ordnung? (Everything all right?)
In case of the broken leg Nein!! would be a plausible answer ;-)
It bears mentioning that in CPR protocol in the United States, you're supposed to begin offering first responder assistance to a collapsed person by shouting, "Annie! Annie! Are you OK?" So for an American English speaker, the most natural/ idiomatic translation of this health-related inquiry is probably "Are you OK?" and not "Are you well?" or "Are you healthy?"
I don't think I'd ever say "Are you well?" unless I'm cosplaying steampunk or at a tea party. "Are you healthy?" is similarly unnatural.
Truthfully, it's an odd sentence in German. They would normally ask "Geht's dir gut?" (literally "Does it go you well?" or translated as "Things going well?") if they were asking if you felt okay because you looked ill or had been ill before.
The only place I can see for this sentence is the idea of "health" as in "Are you healthy? -- Do you exercise regularly? -- Do you eat right?" not "Are you feeling well?"
I could see this question on a survey "On a scale of 1-10, how healthy (gesund) are you?"
I hope that helps give an idea of this usage.
I don't think there is a general rule, but with time patterns will emerge. Vowel shifts usually go from e->i/ie, a->ä, o->ö and u->ü and they can happen for all kinds of reasons, e.g. plural (Apfel -> Äpfel), comparative (gesund -> gesünder), verbs (ich esse -> du isst) and often they appear in words which are somewhat related for example ein Haufen (a pile, noun) -> häufen (to pile up, verb) or offen (open, adjective) --> öffnen (to open, verb).
The important thing is that they only can happen and don't have to. It means that you can not tell if a word will have a vowel shift or not just by looking at it. You will have to pick it up as you go along. But sometimes there are some rules. For verbs, only e->i/ie and a->ä can shift and if you know that a verb shifts it only applies to the 2nd and 3rd person singular and only affects the vowel, the rest of the verb behaves regular. The only exception is a bunch (= 7) of modal verbs. You can read more about them here: http://yourdailygerman.wordpress.com/2012/03/19/learn-german-online-verb-conjugation-2/ And of course sein (to be) which does its own thing in nearly every language.
The hints give "healthy" and "fit" as possible translations, but "Are you fit?" was counted wrong. Any insights? Is the hint misleading or is my response correct? In English it's fine to say "Are you fit?" though it would be more usual to have it as part of a more specific question like "Are you fit to travel?" or "Are you fit to continue?"