This is a rule in modern German Some explanations you can find here: http://blogs.transparent.com/german/prepositions-that-require-the-genitive-cases/ and here http://sok.ch/2009/08/nach-wahrend-genitiv-oder-dativ/
You're right, there is no possession. "Nacht" is genitive here because of the preposition "während." Most prepositions take the dative or accusative case after them, but a few use the genitive (Here's a list). The genitive for prepositions is falling out of use, though, and for most of these prepositions it's acceptable to use the dative instead.
Also, a small correction. It's the word "Nacht" that's genitive, not the sentence. Nouns have cases; sentences don't.
I wish we could use lingots for when we have no internet access, like when traveling and it is not even possible to access the site.
Eh? The streak freeze is still available in the shop for 10 lingots, at least on the website. I'm sure the mobile apps have something similar as well.
Only helps for one day at a time, but still.
Though honestly it's called a "streak" for a reason. I sympathize, but in my opinion it's quite reasonable to allow for only a limited amount of streak saving when the whole point of the streak is that you're doing Duolingo every single day. Sucks to lose it, sure, but I think it's a stretch to still call it a streak when you haven't done it for three days straight.
Actually the article isn't the issue here, but the verb conjugation. We can use "during the night / während der Nacht" to mean either of those.
In English we say "I am drinking" to talk about doing something right now, and "I drink" to talk about doing something habitually (e.g. every night). German doesn't make that distinction in its grammar, so "Ich trinke während der Nacht" can mean either drinking right now tonight, or drinking habitually every night.