Learn Russian: Кроме того, я не люблю эту музыку.

The preposition кроме can mean both "except" and "besides" or "in addition to" depending on the context. When combined with того (the genitive form of то, meaning "that"), it forms a very common fixed phrase used to link sentences, meaning "besides" or "furthermore."

Could I use кроме этого instead of кроме того here?

You could, but it changes the focus. Кроме того is a set transition phrase meaning "besides" or "furthermore." If you say кроме этого, it sounds much more literal, meaning "except for this specific thing" or "in addition to this specific thing" that you were just talking about.

Since the verb is negated (не люблю), why isn't the object in the genitive case (этой музыки) instead of the accusative case (эту музыку)?

This is a great catch! While negated verbs in Russian often force the object into the genitive case (known as the genitive of negation), it is not a strict rule for every verb. With любить, native speakers almost always keep specific, tangible objects in the accusative case (эту музыку), even with a "not" (не) in front of it. The genitive of negation is usually reserved for abstract concepts or to emphasize the total absence of something.