The battery has had it. Try googling for a new one with laptop make/model (
eBay?) The laptop I'm writing this on is mains tethered for the same reason, I'd do well to get 1/2 hour out of it. Mind its a P4 desktop processor and I never got more than 70-80minutes out of it when new.
Wireless network. Again this laptop was always flaky connecting up wirelessly and tended to drop the connection after about an hour, reuiring a reboot to get back onto the wifi. Never had any bother though since I ditched windows XP and installed Ubuntu. Perhaps a bit of an extreme solution, but does point out it was software problem, not the hardware. So, get the latest drivers for your devices.
Have you got anything else that can connect to the network via wifi? PSP, phone, Wii - is it just the laptop or everything?
Problems sound so similar you've not got an Advent 7036 have you?