Emulators are handy because you don't constantly need to transfer all the data to your psp. I fail to see what the PSP firmware can do about that. I also failed to see what you meant with 'homebrew firmware' so I assumed you meant the PSP firmware :P.

There are 2 emulators so far I know, one need the psp to be hooked up (probably only renders images receiving from the psp) and the other one is relatively new. It can´t play commercial games thus well programmed homebrew games should fail too, but simple ones may work.

Although the PSP is much older than the NDS, the homebrew started to grow not so long ago. Perhaps this is the case too with emulators.

About the new PSPE emulator: http://www.emulator-zone.com/doc.php/psp/pspe.html