1 - If you got it when the PSPs first came out, it's a fat (1000) model. If the back isn't flat (the battery cover buldges/curves out a bit), it's a 1000.
2 - If you're on firmware 5.03 or less, you can softmod it. Else, you'll need a pandora battery. You can buy one or turn a normal battery into one (which will make it unusable as a normal battery, only do this if you have a spare). The softmod sticky goes into detail.
3 - Play pirated PSP games... but you can also play PSX/1 games (I'm serious, a sticky here shows you how to convert them for use on the PSP), run homebrew games and programs, watch youtube and various online porn sites (GoTube 1.2), use IRC and instant messengers (AFKIM), cheat at the games (CheatMaster Fusion for PSP games and homebrew, CWCheat POPS for PSX/1 games), take screenshots, stream gameplay video to your PC to record it, and all sorts of stuff... not to mention all the emulators. NES, SNES, Gameboy/Color, GBA, Master System, Game Gear, Genesis, and there's even an N64 emulator (though most games don't run and the ones that do run slow). The PSP's emulation scene is much stronger than the DS's.
4 - Sticky.
5 - Nope. You can do way more than simply pirate stuff like people think, and you'll be able to switch back and forth between official firmware and custom when you want with a pandora battery (later models can't do that).