The machine won't start without the battery
Thats wierd, it should do. Ive got an old r30 (Win2k, 384mb ram, 25gb hdd) and its battery is no longer holding charge so i just dont bother having it attached. It starts fine without it.
Indeed, im surprised your battery still works if its the one it shipped with.
I would say it *may* need a new BIOS battery by the way - i dont know if this would cause the problems you are experiencing, but its an avenue to explore.
Secondly, what BIOS version do you have. Thinkpad BIOSes tend to be buggy, mine would hibernate at all (or scale the CPU speed) when I upgraded to XP, until i flashed the BIOS to a newer version.
R.E: motherboard. Not worth it. It will cost more than a new,better laptop unless you get lucky and manage to find an identical model one on ebay (try and look for a t22 with a broken screen or something, which you could try and can steal a motherboard from)