Questions & Answers about Honum líður mjög illa.
Why does this sentence start with honum (him) instead of hann (he)?
The verb líða (to feel) is what's called an "impersonal verb." In Icelandic, this means the person experiencing the feeling must be in the dative case, not the regular subject (nominative) case. So, we use the dative honum instead of hann. The sentence literally translates to "To him feels very badly."
How would I use the question word hvernig (how) to ask this as a question?
You would ask: Hvernig líður honum? (How does he feel?). Because líða asks about the manner in which someone is feeling, we use hvernig (how) to form the question. Notice that the person feeling still stays in the dative case (honum) when you turn it into a question.
Why do we use illa for "bad" here instead of a normal adjective like slæmur or vondur?
The verb líða always pairs with an adverb (a word that describes an action) rather than an adjective (a word that describes a noun). Illa is the adverb form, meaning "badly." In Icelandic, you don't feel "bad," you feel "badly." If you used an adjective, it would sound like you are saying "He is an evil person" rather than "He feels sick/sad."
Does the word mjög (very) change if we change the subject to "her" or "them"?
No, mjög is an adverb, and adverbs in Icelandic never change their endings for gender, number, or case. It is completely uninflected, so it will always stay exactly like this.