Learn Swedish: Det är en hemlighet vilken pseudonym som döljer signaturen. - It is a secret which pseudonym hides the signature.

This highlights a mandatory rule for indirect questions in Swedish. Whenever the interrogative word or phrase (in this case, vilken pseudonym) acts as the subject of the embedded clause, you must insert som right after it. Because the pseudonym is the thing doing the hiding here, it is the subject, making som required.

If the sentence were instead "It is a secret which pseudonym the author hides", would we still use som?

No, you would drop it. In that hypothetical sentence, "the author" (författaren) would be the subject doing the hiding, and vilken pseudonym would be the object. You only insert som when the question phrase is the subject. So the object version would simply be: Det är en hemlighet vilken pseudonym författaren döljer.

Why is the verb döljer used here instead of gömmer?

Both verbs mean "to hide", but they are used in different contexts. Att gömma is generally used for physically hiding objects or people in a specific place (like hiding a key). Att dölja is used for concealing facts, information, feelings, or identities. Since a pseudonym conceals an identity, döljer is the natural choice.

Does signaturen just mean a literal, handwritten signature here?