Yep, your battery seems dead, then. I'd try taking it out, leaving it for a day or two and then put it in again, just to see whether it comes back alive. But if it's been deteriorating for so long, it might well just be dead. Replacements from newertech or fastmac should help.
As for the low disk space: Yeah, that _really_ makes a change in performance. Keep 2-3 GB free and it's much better already. There's some percentage rule floating around, but I'd simply keep as _much_ free as possible.
Mac user since 1987. Running Mac OS X 10.8 Mountain Lion on a MacBook Air 11" & an iMac 27" and whatever's newest for my iPhone 4s, iPad 3 and AppleTV 2.
Apple Certified System Administrator 10.6, Apple Sales Professional 2008-2011, Apple Certified Mac Technician.