Up the Tigers
I assume when an enemy moves into an unexplored tile, they flip it face up?
Yes the enemies reveal these and they are placed in the same way as if you revealed them, so you can decide the best way to suit you.
I was a little unclear on how to orient tiles face up when explored. Can I orient them anyway I choose? As long as there is an open space where I moved the character?
Yes anyway you like, as long as the square on the new tile next to where the character is moving from is a valid entry point to the new tile.