Learn Polish: Oficjalny list kończę słowami: Z poważaniem. - I end an official letter with the words: Yours faithfully.

It actually is in the accusative case! Because list is an inanimate masculine noun, its accusative form is identical to its nominative form. It's easy to forget this at higher levels and overcorrect, but only animate masculine nouns (like people and animals) take the -ego and -a endings in the accusative.

Why is the word for "words" in the instrumental case (słowami)?

The verb kończyć (to end) takes the instrumental case when you are stating what you use to bring something to a close. You can think of it as ending the letter "by means of" these words. You will see this pattern often, like in "kończyć się sukcesem" (to end in success).

What is the literal translation of "Z poważaniem" and how exactly does its grammar work?

It literally translates to "With respect" or "With esteem". The word poważanie (respect) is a neuter noun, and here it takes the instrumental ending -em because it follows the preposition z (meaning "with" in this context). It is the absolute standard sign-off for formal correspondence in Polish.

Could I use the perfective verb "skończę" in this sentence?

Not if you want to express a general rule. The imperfective verb kończę is used here because it describes a habit or standard practice (how you always end official letters). You would only use the perfective skończę if you were talking about finishing one specific letter right now or in the future.