Beat the game on Normal difficulty to unlock the Single Level mode.
The Replay option seems to just play a "movie" of the last time I played a particular level. Is there a way to actually go play a specific level? Or do I have to play all the way through the whole game again?
what happens when you beat the game on hard
When you beat Hard mode, you'll unlock Ludicrous mode, in which you have only 1 live, no continues and the game will be on Turbo mode... It's pretty hard, I still haven't beat it myself yet 
Fortunately, there are some shortcuts to skip a few levels...