Is there a way to set initial state with rigid bodies? Let's say I set up a Rigid Body simulation in Blender (not in GE), and then run the simulation for a number of frames. Is there a way to tell Blender to use the current state and position as the initial state? So that when I go back to frame 1, the rigid bodies stay in the same place they did in frame X.

For instance, if I want a pile of bricks, I first place the bricks in the air and then let them drop down on the ground, coming to rest. When they have come to rest, I want to use that position as the starting position.
Use the "Apply Visual Transform" operator. You can find it in the "Apply" submenu of the "Object" menu in the 3D view header or by pressing ctrl+a.
This way you don't need to bake the simulation and don't loose the rigid body settings.