Learn Turkish: Kedi içeri giriyor. - The cat is going inside.

This is a special exception! Words that already express a pure direction, like içeri (inside), dışarı (outside), yukarı (up), and aşağı (down), do not need the 'to' suffix (-e or -ye) when used right next to a verb of motion. You can say içeriye, but just içeri is more common and perfectly correct.

Which question word would I use to ask this sentence as a question?

Because the cat is moving toward a destination, you would use Nereye (Where to). The question would be Kedi nereye giriyor? (Where is the cat going in?). You would only use Nerede (Where at) if the cat was already sitting still inside.

The English translation says 'going', so why is the verb giriyor instead of gidiyor?

Giriyor comes from the verb girmek (to enter). In Turkish, it is much more natural to pair içeri (inside) with girmek (to enter) rather than gitmek (to go). So, 'going inside' is expressed literally as 'entering inside'.