Usually we don't include "over" in this - either "across" or no preposition, in my recollection.
Yes, in proper place names that consist of more than one word, we only capitalize the first word. In this case it happens to be the definite article.
Thank you. [Tweaking my hard wiring to make sure I got that this time.] ;-)
Yes. You'll see this with a lot of proper nouns, particularly the ones that entered the language back when the Danish influence was stronger. "Det hvite hus", "Den europeiske union", "De forente stater", "Den norske kirke"...
In which cases do I need only one article? I mean, does the rule deal only with special objects (such as organizations/countries in your examples)? Are there any other cases/examples?
Kanskje stiller jeg spørsmålet på en unkorrekt måte. Jeg vil gjerne vite nå man trenger bare en artikkel (den, det, de), ikke ending (-en, -etm -ene).
Det eneste jeg har sett er eksempler med organisasjonens/lands navn.
It's generally limited to proper nouns, which is why you're seeing countries and organizations. Book and film titles are often affected, statues and buildings with names can be too: "Den tredje mann", "Det hvite hus".
In formal/academic language you'll sometimes see the definite ending being dropped even for regular nouns, but that's not something I'd recommend experimenting with as a learner, nor is it something we accept in this course.