Hmm, this sounds a bit like the behavior my N900 started doing when my battery was going bad (although I usually got more than two minutes out of it). My first battery lasted a bit less than two years if I recall correctly, so they don't necessarily last all that long. They are fairly cheap to replace, though.

Again, my first guess would be a battery problem; that might explain why the phone works correctly while drawing power from the USB port. And yeah, if you choose either "mass storage mode" or "pc suite mode" when you plug in the USB cable, your memory honestly isn't available -- the internal storage is unmounted from the Maemo OS and made available to the host computer as a USB drive. You'd need to click in the blurred area above the dialog box to avoid this behavior...