Learn Polish: Po co gromadzisz te rzeczy? - What are you hoarding these things for?

Po co asks about the purpose or goal ("what for"), while dlaczego asks about the reason or cause ("why"). By asking Po co gromadzisz te rzeczy?, the speaker is focusing on what you actually plan to do with all this stuff, rather than asking for the psychological reason behind your hoarding.

Could I use czemu instead of po co in this sentence?

You could, but it changes the nuance. Czemu is a more colloquial version of dlaczego (why/for what reason). If you say Czemu gromadzisz te rzeczy?, you are asking what caused them to start doing it in the first place, rather than what their end goal is (po co).

What case is te rzeczy in this sentence?

It is in the accusative case, because it is the direct object of the verb gromadzić (to gather/hoard). For non-masculine-personal plural nouns like rzeczy (things), the accusative form is identical to the nominative (dictionary) plural form. The demonstrative pronoun ta (this) becomes te to match.

Why is the verb gromadzisz in the imperfective aspect here?

Hoarding or gathering is typically an ongoing, continuous process. The imperfective aspect shows that the person is currently in the habit of accumulating these things, rather than asking about a single, already-completed action.