Learn Russian: Мы получаем результаты после проверки.

It follows the preposition после (after), which always requires the genitive case. The dictionary form is the feminine noun проверка (inspection). Normally, feminine nouns ending in -а change to -ы in the genitive, but due to the Russian spelling rule that forbids writing ы after к, it changes to -и instead.

Is результаты in the nominative or accusative case here?

It is in the accusative case because it is the direct object of the sentence (the thing being received). However, for inanimate nouns like результат (result), the accusative plural is identical to the nominative plural.

Does получаем mean we are getting the results right now, or that we get them regularly?

It can be both. Получаем is from the imperfective verb получать. The Russian present tense handles both continuous actions (we are getting) and routine actions (we usually get). The context of the conversation determines which one is meant.

What exactly does проверка mean in this sentence?

Проверка comes from the verb проверять (to check). It broadly means a check, verification, or inspection. Depending on the context, this sentence could mean getting results after a medical check-up, an audit, or a security inspection.