There is no way to manually save, and you are stuck in the save slot that you are playing in. There are auto-saves at the beginning and end of every boss fight (and, mercifully, half-way through one of them) and they come up frequently throughout the Story Mode. There is an onscreen cue to let you know when the game is saving.
If I kill a clickety-clacker or find one of the Riddler's question marks, it does save yes.
But let's say I go through an area and beat bad guys or maybe decontaminate a deadly toxin. If I happen to quit the game or turn off the system right away, it takes me back. And yes, I do wait for that save icon to finish saving before I try anything.
Yeah, if you want to save or if you did something in a room, just go back to the room you came in if you can. You'll see the swirly little icon as it saves your game. Obviously sometimes this isn't an option but that's rare.
"I'm counting on you, ultimate sword" - Jaster Rogue