Learn Polish: Widzę pierścionek, który wystaje spod dywanu. - I see a ring that sticks out from under the carpet.

Spod means "from under". It is a compound preposition made by combining z (from/out of) and pod (under). Just like z on its own, these compound prepositions of origin always require the following noun to be in the Genitive case, which is why we see dywanu.

Why is pierścionek not in the Accusative case, since it is the direct object of widzę (I see)?

It actually is in the Accusative case! Because pierścionek (ring) is a masculine inanimate noun, its Accusative form is identical to its dictionary (Nominative) form. If it were a masculine animate noun, like pies (dog), its ending would change (for example, Widzę psa).

The English translation uses "that". Why does the Polish sentence use który instead of że?

In English, "that" can be a conjunction (I know that you are here) or a relative pronoun pointing back to a noun (the ring that sticks out). In Polish, you must use że for the conjunction and a form of który for the relative pronoun. Since "that" refers back to the masculine noun pierścionek, we use który.

Why does the Genitive form of dywan end in -u (dywanu) instead of -a?