"John is not well: he is feeling dizzy."
Translation:János nincs jól: szédül!
Now we are translating names too? This is not consistent across the exercises. For example, Zsuzsa remains Zsuzsa, not Susan.
Why is "János nincs jól: ő szédül" wrong? What is wrong with using "Ő" here?
It's grammatically correct, but unnecessarily emphatic. It's clear that it's János who is feeling dizzy, there's no need to emphasise it with the pronoun. But as I said, it's not incorrect, so they might add it as a possible translation if you report.
It depends on the context. You're right in that it often means a lack of something, eg "nincs pénz" means "there is no money". However, Hungarian words and phrases don't necessarily have a 1-to-1 translation to English. In this context, you can think of it as a lack of well-being.