"Now, it is snowing inside."
Translation:Artık içeriye kar yağıyor.
Artık means "now" but in a way that is associated with a change, reversal, or new condition. It divides time between before, when things were one way, and after, when they are another. So in this case: "[Formerly it was only snowing inside, but] now it is snowing inside."
The other translations you offer could easily apply in other sentences with artık.
"Ben o kitabı artık okumuyorum" - [Formerly I was reading it, but] I am not reading that book anymore
"Sen artık genç değilsin" - [You once were, but now] you are no longer young
"yağıyor" (whether raining or snowing) can use locative (at/in/on) or dative (to) on the noun. Locative is from the static perspective of the ground, I guess... it's raining here, there, wherever. Dative subtly implies movement... it's raining/snowing from the sky to the place.
In this situation, I'd personally lean more towards "içeriye" because it's kind of weird that the snow is coming into the building. So I would want to emphasize the movement, rather than simply stating it's snowing at a certain place. But they are interchangeable, and Duo should accept either way.
Here are a few more Duo sentences to compare, with links to discussions about them- where native speakers (in particular: Ektoraskan, a former contributor) said we can use either case: