What is the difference between illa and dålig? Is illa more used for food?
"Illa" is an adverb, "dålig" is an adjective. Illa means badly/poorly/not well. It's synonymous with the adverb "dåligt".
Is there a way to distinguish "she's having a bad day" from "she has a bad day" in Swedish? If I wanted to stress the fact that it's an ongoing experience?
does "har" correspond in general to English "have"? eg. in the senses of "we had a good time", "they're having an argument", "I have to leave soon"
I can't answer your question definitively because my Swedish is still infantile, but roughly 'att ha' (present tense: har) corresponds nicely with English 'to have'.
However that doesn't mean it translates to 'have' in every instances of 'have' in English, because you have to be aware that most of the time, 'have' in English are used idiomatically. For example, in your instances, 'have a good time' and 'have an argument' are idiomatic expressions, and there are possibilities that in Swedish those could very well be translated into 'make/share/any other potential verb a good time' or 'do/make an argument'. Wait for a Swedish native to tell you the suitable translation for those expressions.
'to have to' on the other hand, does not mean 'to possess' in English. It means 'to need', expressing obligation. So I'm pretty sure the equivalent in Swedish is not a one-to-one correspondence. But I might be wrong, considering the close linguistic relationship between Swedish and English.