Learn Russian: Хотите выпить чашечку кофе?

This is a classic example of a softening diminutive. The speaker isn't necessarily offering a physically tiny cup. Instead, adding the -ечк- suffix makes the offer sound warmer, more hospitable, and polite. You will hear this very often in Russian when hosts offer food or drink.

Why is the perfective verb выпить used here rather than the imperfective пить?

When offering a specific, single serving of a drink, you use the perfective aspect because the focus is on the result of completing that single action. Using пить would sound strange here, as it implies the general process or habit of drinking.

Why does the ending of чашечка change to чашечку in this sentence?

It is in the accusative case because it acts as the direct object of the verb выпить (to drink). The feminine diminutive ending -а changes to -у just like a regular feminine noun.

What case is кофе in here, and why does it not change its ending to show possession or quantity?

It is functionally in the genitive case because it means a cup 'of' coffee. However, кофе is a foreign borrowed word and is completely indeclinable in Russian. Its form never changes no matter what grammatical case it is in.