Learn Polish: Wirus pozbawił mnie moich plików. - The virus deprived me of my files.

The phrase moich plików (my files) is in the Genitive. The verb pozbawił (deprived) requires the thing that is taken away to be in the Genitive case.

What case is mnie here? Is it also Genitive?

Actually, mnie is in the Accusative case here, even though it shares the exact same form as the Genitive. The grammatical structure is pozbawić kogoś (Accusative person) czegoś (Genitive thing). You are directly acting upon the person (Accusative) by taking away their things (Genitive).

I often see short pronouns used in the middle of a sentence. Why isn't a short form used instead of mnie here?

While pronouns like ty have a short Accusative form (cię) used mid-sentence, the short Accusative form for ja (mię) is archaic. In modern Polish, you always use mnie for the Accusative, regardless of its position in the sentence.

If the virus were currently in the process of deleting the files, how would this sentence change?

You would use the imperfective verb pozbawiać instead of the perfective pozbawić. The sentence would become Wirus pozbawia mnie moich plików (The virus is depriving me of my files).