The course appears to be organized and everything is introduced very clearly in the tips and notes. Do "dobrý" and "den" mean different things on their own, or is it just one word in two parts, sort of like the English "By the way" - each of the words in it is a different word, but the whole thing translates differently?
Yes, people do. https://www.ceskatelevize.cz/porady/10435049455-dobre-rano/
It is used in the early morning and only once though, not the whole time before noon like in English. 11AM is the time for Dobré dopoledne, not for dobré ráno, unless they are sarcastic and pointing out to the fact that you just got from your bed so late.
It is unlikely other tourists would greet you like that in the mountains. Only at the breakfast or when packing your tent or really early in the morning.
"Good day" is not very common in English (apart from the Australian g'day) but I think it really would be better to translate it literally. But I think that for example the Spanish and the Portuguese Duolingo courses also translate "buenos dias" and "bom dia" as "good morning".