It's pretty widely used, especially in some combinations like vilken tid på dygnet ('what time of day'), dygnet runt (= 'around the clock'), dygnsrytm (hard to say in English, but maybe 'sleep pattern', it refers to when you sleep and when you're awake), femdygnsprognosen on TV which is the five-day weather forecast etc etc, so I think this word is hard to miss if you grow up in Sweden.
Yes, Swedes sometimes tend to misuse för in contexts like this, influenced by English I guess, though there may also be some kind of dialect/historical reasons that make it possible. Schools aren't really interested in teaching normative grammar and people hear and read a lot of English, so it happens pretty often. (this is my personal opinion)
This made me think of the poetry book Varför har nätterna inga namn? ('Why don't the nights have names?') by Göran Palm. In the title poem he wonders why only the days have names, there's only 'tisdag', but no 'tisnatt' to go with that. I have to agree with him, it is a bit unfair!
It’s always used in like scientific contexts, and you can’t really say that Earth rotates one lap a dag, then it would be incorrect. It’s also often used if you want to stress that something happened during the night as well as during the day. Normally, you sleep at night, so there’s less focus on that. But there is a younger colloquial word called dygna which is ’pull an all-nighter’. Or if someone has a job like a doctor and they forced to work both during the day and during the night I could describe it as having worked ett dygn.
That is, when it’s used in normal speech, at least in my experience, it’s often to show that something happened both day and night. If there were only activities during the day and everyone slept safe and sound at night, you might as well use dag.
I am now starting the final lesson of this chapter and still make a lot of mistakes. For such a difficult subject as knowing what words to use during specific situations concerning location and time, this chapter is way too short. To top it off there is no explanation we can learn from. We basically have to guess our way through this very important chapter and as such learn nothing. Can you please improve this?