Homebrewing is similar to rooting. It allows you to install stuff that isn't on the phone, like various apps to add functionality. I really don't know much on details because I have little interest in it. The only thing on Homebrew that appeals to me is the ability to run various Emulators to play old NES, SNES, and Genesis games, but I picked up a little portable gaming system on ThinkGeek that does that out of the box for pretty cheap that I don't have to worry about bricking my PSP. I'm sure there are many many many things out there for homebrew info, but I'm not really into the whole thing, so I can't speak for more than emulation.
Also, Sony updates the firmware on their PSP fairly regularly. While they do add some functionality that can be welcomed, it also locks out any homebrew stuff. Where they get you is when future games require a certain firmware to run and you can either update your firmware and lose homebrew, or not buy new games.
Online gaming really depends on the game. I haven't really played any that are heavy online games, so I can't really comment on how well it works.
The graphics on the system are quite good. They're crisp, the screen looks beautiful, and it's really the best part of the system. The system has been out long enough that most of the games look quite impressive.
Dead Pixels: my first PSP had one. Not actually a dead pixel, but a stuck pixel. It was mildly annoying to have a little blue dot that wouldn't change, but it wasn't enough for me to do a warranty swap. My 3000 has been flawless. I like the disc door better on the original PSP, but I understand completely why they simplified it. Quality on the unit is about top notch. No bad edges or goofy buttons, though the one analog stick that is on it is just awful. It's too small, it doesn't feel right, and it makes my left hand cramp up like crazy from being in an awkward area. If it was about half an inch more to the right, it would make things much better.