Learn Portuguese (Portugal): Cada álbum tem uma capa diferente.

No, cada (each/every) is an invariable word. This means its spelling never changes, no matter the gender of the noun it describes. You say cada álbum (masculine) and cada capa (feminine).

Since this lesson covers both cada and qualquer, could I use qualquer in this sentence instead?

Using qualquer would change the meaning to "Any album has a different cover." Cada specifically means "each" or "every (single one)," which makes much more sense here because you are pointing out that each specific album has its own unique cover.

What is the plural of the word álbum?

The plural is álbuns. As a general rule in Portuguese, nouns that end in -m change to -ns when they become plural (just like um homem becomes dois homens).

Does capa really mean "cover"? I thought it meant "cape" or "cloak".

It means both! Capa is the word for a clothing cape, but it is also the standard, everyday word for the cover of an album, book, or magazine. It is even used for a mobile phone case (capa de telemóvel).