Learn Polish: Czyje jest to biurko?

The question word for "whose" acts like an adjective in Polish, so it has to match the gender of the noun it asks about. Since biurko (desk) is a neuter noun because it ends in -o, we use the neuter form czyje. If you were asking about a masculine noun like pies (dog), you'd use czyj, and for a feminine noun like książka (book), you'd use czyja.

Since this lesson is about the Genitive case for possession, why aren't any words in the Genitive case here?

In this question, biurko (desk) is the subject, so it stays in the basic Nominative case. You are asking about the owner, rather than naming them. However, the answer to this question will use the Genitive case to show who the owner is! For example: To jest biurko brata (This is the brother's desk), where brata is the Genitive form of brat.

Could I just say Czyje to biurko? without the word jest?

Yes! In casual spoken Polish, the verb jest (is) is often dropped in simple questions like this. Czyje to biurko? sounds very natural and means exactly the same thing.

Can I change the word order to Czyje to jest biurko?