When one says, "I took my hat and my leave", that can be identified as a "rhetorical device" and labelled a "zeugma". Unusual uses of language draw attention, create emphasis, and what's unusual here is the use of the one word in a literal and a metaphorical sense.
I suspect the same thing is going on with "har hoste og vondt". I think one "har" en hoste in a different way from how one "har vondt".
Anyway, so one thing that the sentence teaches is that zeugmas work in Norwegian.
I'm a big fan of zeugmas and applauding you for bringing them up :)
I just realised English with its huge number of phrasal verbs should be perfect for them - "after a heavy night of drinking, I get up late and a headache".
With "get", you could probably create arbitrarily long chains... "When we drink, we get along well together, but the next day up late, a headache and sometimes fired".
Another way to say this is" Jeg hoster og har sår tunge" The doctor is asking what your condition is, and you can answer with that. Legen spør: Nå, min venn, hva feiler det deg i dag? ( Now my friend, what is " wrong" with you today) and you answer: Jeg hoster og har sår tunge ( I am caughing and my tongue is sore.)
Not quite sure what you mean by "Norwegian words that have an 'ø' but that sometimes appear as an 'o'"?
Are you thinking of vowel changing nouns/verbs, like
bok - bøker
å gjøre - gjør - gjorde - har gjort
In that case, I'm afraid I can't immediately think of any tips apart from "learning the words", which isn't much help.
The word itself will definitely stay the same, so if you know that the word for books = bøker, you are all set! It's not going to change to "boker" in certain cases.
As you might know, the "proper" way to spell an ø with an English keyboard, is oe (ae for æ and aa for å), but since a lot of people here (and elsewhere) tend to just spell it using a simple o, it can probably get a bit confusing.
So sorry if I'm completely misunderstanding what you're asking, as I'm not the best at reading/interpreting comments.