Am I understanding things correctly, that I can run a dynamics simulation and save my hair prop at a given frame in such a way that it will load in that position by default? If so, what's the easiest way to accomplish that?

Just export it as obj while standing in the frame you want, or you can choose it when it ask if you want a single or range of frames. Just find it easier to just be in the correct frame. After you have exported it you just import it again. 

If you have "special" (Meaning more than just diffuse) material on the hair its a good idea to just save it to the library as it will most likely not be correct after you import it. So its just so its easier :)

Okay, scenario.

If I like frame 7, and save it to the library on frame 7, will it be in that position on frame 1 when I reload it?

I don't think exporting dynamic hair will work. Cage has a script that will bake dynamics into hair (and clothing I think) here http://www.renderosity.com/mod/forumpro/showthread.php?thread_id=2839086&page=3#message_3861962 But a word of warnig. The skull cap must be in a 0,0,0 position when it's not parented to a figure. having it zeroed while parented or having the firgure zeroed won't help. The hair group will fly off. You need to run the script with the hair group selected and run it for each group. You can them use the animation palate to move the key frame that is created around.

Sorry i misread now that i saw Reds answer, i think he is right. So just ignore my first post :D