I know that the Nintendo DS is 802.11g compatible - does this mean that I can use it to get on the wireless network in my house and chat/deathmatch/dick around? Or does it simply mean that if someone else is using a DS within 50 feet of me, I can hook up to them?
I’m pretty sure both, but at the moment they don’t have the setup for over the internet play yet. Supposedly when they do, though, you should be able to use a wireless internet connection to play people anywhere in the world. There was also talk of using the DS as a phone with VoIP (voice over IP) but that’s just rumour so far.
Yes, as above; for the latter the DS has its own proprietory wireless interface that uses less power. One of the plus points of the DS over the PSP, which only has the power hungry 802.11. However, the kind of things you can do with both consoles are so different that you’ll easily decide which one suits you best. Basically, if you liked the GameCube better than the PS2 (for reasons of available games rather than graphical prowess, mind you) then the DS is your handheld, and if you liked the PS2, including its capacity to play CDs, DVDs and so on, then the PSP is your handheld (with its beautiful and very large 16:9 display and mp4, mp3, and jpg support - I own this one, and it’s my personal favorite 
although some minigames, like the catapult one in Mario 64 DS which you play with the pen and touchscreen, is quite funny).