It's one of the alternatives, so it should have been accepted, even if not very natural. Perhaps there was a glitch, or another typo at play.
The 'my' is sometimes implied in Norwegian when it's rather obvious that the person or thing which you are referring to is in fact yours. It is most common in answers to questions about that person or thing, where the context of that person or thing being yours is already established:
Misliker du svigerfaren din?
Do you dislike your father-in-law?
Nei, tvert imot, jeg elsker svigerfar! Han tar alltid med kake og øl.
No, quite the opposite, I love my father-in-law! He always brings beer and cake.
However the possessives never -have- to be omitted, so as learners I'd advice you to add them as a general rule. You're of course safe to omit them in the sentences where the course does so.
I always felt (in English) that using it without the possessive feels like talking about the person, whereas using the possessive feels like talking about that person in that role.
Dad is cool = My dad is, in general, a cool person. My dad is cool = My dad is cool as a father.
Would the same be true for Far er kul / Min far er kul?
Well our teacher told us it doesn't really matter if you place the possesive in front or after the subject, but that if it is in the front, it sounds very childish to her (f.x. a child is screaming: "min bil, min bil"! - my car, my car! and adult will rather say "bil min").
This depends as much on tone of voice as on word placement, so I'd take that advice with a grain of salt. It does however tend to put focus on the ownership aspect.
When describing family members that rule goes out the window, as we don't own family. Saying 'min svigerfar' is actually much more common than 'svigerfaren min', though either is fine.
Well it pretty much depends on the context I would say. For example it makes sends and doesn't sound unnatural at all if I say "I love father-in-law" in Czech language without a possesive. I admit it sounds a bit strange in english, but technically I would say it is correct.
Even in Czech I would say "miluji svého tchána" rather than "miluji tchána", but Czech is in any case an unreliable guide to English in these matters. English countable nouns generally require determiners, at least in the singular. These are often articles, but can belong to other categories, including possessives. That's why English translations of Czech sentences often include possessives where Czech has none. English requires a determiner and prefers possessives to articles where both are possible. Czech has no articles and no aversion to countable nouns without determiners. So I would say that "miluji tchána" is grammatically correct Czech, though a bit weird, while "I love father-in-law" is not just weird in English but grammatically wrong.