You can't do much about the issue of context. But I would suggest that "Is he unwell?" isn't a good translation of this sentence. If you already knew that he was unwell, you could ask an bhfuil sé go dona to find out just how unwell he is, but if Joe didn't turn up for work today, and you wanted to know if he was ill or just off playing golf, you wouldn't ask an bhfuil sé go dona.
But go dona doesn't actually mean "unwell" or even "poorly" - go dona in a health context usually implies something more serious, just as you wouldn't ask "is it bad?" when the choice is between a hiccup or a sneeze. "Is it bad" tends to be used when you mean "Is it going to hurt a lot?" "Will there be a scar?" or "Am I going to die?" not when you mean "Does this mean that I won't feel like eating my dinner?"
(Yes, I know that you can have a "bad papercut", but that is usually reserved for one that bleeds a lot, or is going to sting).
You can do a job poorly, or badly, or not do it well and you would use go dona, but I think it might be a bit of a béarlachas to stretch that to "unwell".
Your second question highlights this - I don't think you'd ask an bhfuil tú go dona?, you'd say an bhfuil tú breoite or an bhfuil tú tinn?
Too bad we don't have context to know which of those two situations it would be. ;) But the other issue, aside from context, is what translations (for English) are acceptable for the word itself; I'd use "unwell" before "poorly", regardless of whether asking the question at all is appropriate.
To re-align a little bit, would it be better in second person? Is looking at someone in front of you and saying an bhfuil tú go dona more plausible?
In case anyone else was wondering, "dependent" verb forms are a particularity of Celtic languages: "In the Goidelic languages, dependent and independent verb forms are distinct verb forms; each tense of each verb exists in both forms. Verbs are often preceded by a particle which marks negation, or a question, or has some other force. The dependent verb forms are used after a particle, while independent forms are used when the verb is not subject to a particle." https://en.m.wikipedia.org/wiki/Dependent_and_independent_verb_forms
The problem with quoting whole chunks of things that you don't fully understand is that they often leave out important information that will confuse other learners. In this case, Wikipedia doesn't point out that, except for some of the irregular verbs, the dependent form of the verb is the same as the independent form. For most learners, the difference is more theoretical than practical, as it really boils down to irregular verbs are irregular.
I am a beginner, but translated this to mean "Is he feeling badly?" Is that not an acceptable translation? (obviously not in this lesson) I know it is a switch from adjective to adverb, but is that not what the question means, assuming the subject is he rather than it? One of the listed meanings in the pull down is "feeling bad"