Learn Hardcore Russian: Оплаченный счёт лежит в папке на столе. - The paid bill is in the folder on the table.

Оплаченный is a past passive participle from the verb оплатить (to pay). In this sentence it behaves like an adjective and agrees with the noun:

оплаченный (masc. nom. sg.) + счёт (masc. nom. sg.) Meaning: the paid bill/invoice.

Why is it оплаченный счёт and not оплачен счёт?

Russian has two common participle/adjective-style options:

оплаченный счёт = “the paid bill” (long form; describes the noun like an adjective; very common in writing and speech)
счёт оплачен / счёт был оплачен = “the bill is/was paid” (short form used predicatively, like a statement about the bill)

Your sentence uses the attributive version (оплаченный) to label which bill it is.

What case is счёт in, and why?

Счёт is nominative singular because it is the subject of лежит (“lies/is lying”):

(Что?) счёт лежит = “(What?) the bill is lying…”

What does счёт mean here: “account” or “bill”?