Learn Russian: Почему ты так медленно подходишь к двери? - Why are you walking to the door so slowly?

The preposition к means "towards" or "up to" the outside of something. Because you are approaching the door but not going inside it, you use к, which requires the Dative case.

Why does the word for door end in -и (двери) instead of -е for the Dative case?

Most feminine nouns end in -а or -я and change to -е in the Dative case. However, дверь is a feminine noun that ends in a soft sign (-ь). These types of nouns drop the soft sign and add -и in the Dative case.

How does the verb подходишь relate to the preposition к?

The verb подходишь is made of the prefix под- (approaching) and ходить (to walk). Verbs of motion with the под- prefix almost always pair with the preposition к because they both express moving "towards" or "up to" a target.

Why is it так медленно and not такой медленно?

медленно is an adverb meaning "slowly." When you want to say "so" before an adverb, you must use так. You only use такой before a noun or an adjective, for example, if you were saying "such a slow door."