Learn Hardcore Russian: После ужина я попросил счёт у официанта. - After dinner I asked for the bill from the waiter.

счёт is in the accusative case because it is the direct object of попросил. Inanimate masculine nouns have an accusative form identical to the nominative.

What does у официанта mean and what case is официанта in?

The preposition у takes the genitive case to express possession or source. официанта is the genitive singular of официант, so у официанта means from the waiter.

What’s the difference between просить and попросить, and why is попросил perfective used here?

просить is imperfective, for ongoing or repeated requests
попросить is perfective, for a single, completed request
Since this sentence refers to a one-time completed action, the perfective past tense попросил is used.

Is the pronoun я necessary in this sentence?

No. Russian often omits the subject pronoun when the verb ending indicates the subject. You can say После ужина попросил счёт у официанта and it still means I asked for the bill after dinner.

Can I change the word order, for example putting после ужина at the end?

Yes. Russian has flexible word order. For example:

Я попросил счёт у официанта после ужина.
Я попросил у официанта счёт после ужина.
Both convey the same meaning; changing position shifts emphasis slightly.

How do you pronounce счёт, and why is it written with ё?

счёт is pronounced [ɕˈot]. The cluster сч merges into a long palatalized [ɕː] sound, and the letter ё denotes the stressed vowel /o/. Writing ё instead of е clarifies both pronunciation and stress.

What’s the difference between попросить счёт and принести счёт?

попросить счёт means to ask for the bill (customer’s action)
принести счёт means to bring the bill (waiter’s action)