I've always wondered about this scenario. Is this checkmate?

Can the black King take the rook on E8 (the rook on E2 cannot move because of the Queen on A2)?

Yes, this is checkmate. The king is not allowed to put himself in check. You could rationalize why this would be the case by thinking what would be white's move after the black king would hypothetically take the rook. White would take black's king and is faster than black. And we all know from the last scene in Lord of the Rings that when your ruler gets killed, it doesn't matter anymore whether you're completely surrounded and outnumbered cause they'll just run away. 

Yes, that is mate. Even if black were to hypothetically take the rook, the black king would be captured before the white king. Thus white wins cause he has the last king standing.