Once I am in a game can I see what difficulty it is set to in any way without having to go kill something and see what level loot it drops?
UPDATE: As of Diablo 2.0, this is no longer the case. The difficulty can be seen in the top right of the screen, next to the clock, while playing.
The only way I'm aware of is at the character select screen:
You can change the difficulty (assuming you've unlocked them) by going to the Select Quest screen, and changing the difficulty of the quest you're going to start.