Correct. This sentence can't stand alone, but unfortunately, we have no idea what is the question that we're answering. It can be 'how are you' 'How is this or that going' 'how did the meeting go' 'how is he/she' 'have you travelled well(comfortably)', etc.
This is just a general expression to say 'not well', there's no verb, no subject, no context. It's just a short answer.
How do we know "he/she is" is not understood in this sentence? Does it depend on context?
If we ask about him/her, then yes, we say 'Nincs jól'. Note that 'nincs' is a verb, and in this sentence, there's no verb, and there isn't any indication about who/what we're speaking about. It's just an expression, without any context. It's just the denial of one word.
I am hearing "SHOY-nos" for sajnos, and wondering if that is the correct pronunciation.
I put I'm sorry I'm not well and sajnos means sorry so am I correct?
I got this wrong too. There's no subject or verb in the Hungarian prompt. Some of the earlier comments point out that this prompt can't stand on its own and needs context to make any sense. That said, I don't think the first person verb can be omitted. If there were a subject, she or the boy or the children, the verb could be left out. As it is, it's not a sentence and just frustrating for a Hungarian learner.