Learn Russian: Вы можете попросить кого-нибудь принести кофе?

The word кто-нибудь is the base (nominative) form. In this sentence, "someone" is the direct object of the verb попросить (to ask), so it needs to be in the accusative case. For animate words representing people or animals, the accusative takes the same form as the genitive case, turning кто into кого.

Why use попросить here instead of спросить? Both translate to "ask".

спросить means to ask a question to get information. попросить means to ask for a favor or to request that someone do something. Since you are asking someone to perform an action (bringing coffee), you must use попросить.

Why are the verbs попросить and принести in the perfective aspect here?

You are asking for a specific, one-time action to be completed with a successful result (asking someone, and them bringing the coffee). Perfective verbs are used for these single, completed events rather than ongoing or repeated habits.

Why doesn't the word кофе change its ending here?

кофе is a borrowed foreign word and is completely indeclinable in Russian. It never changes its ending, regardless of its grammatical case. Even though it is the direct object here (accusative case), it remains exactly the same.