PSP 1000 - Fully hackable
PSP 2000 - Some are some are not
PSP 3000 - None are fully

If you have a 2000 (some) or a 3000 you have to temporary hack it (meaning when you turn the handheld off you have to redo it again. Sleep mode is what you would keep these in at all times) but it's not hard by any means, just make sure you download the right files and put them in the right places.

http://wololo.net/cfw4dummies/ (Great guide to follow)
http://www.youtube.com/watch?v=N1J9YsW93FE (Great video to follow)

As for emulation don't expect to be playing anything N64 and up in terms of power. So anything PS1 and down (besides Saturn) i'm fairly sure you could play with no problem.

Hack once,

Fast recovery any other time. Don't keep it on/sleeping all the time.

Whats the best homebrews?

My personal favorite:

http://www.youtube.com/watch?v=-Y8fKu-u4Is