I can tell you from experience that when the battery dies the car will not run. The symptons of a 2001 - 2003 Prius that has a battery that is really dying is the car goes into neutral with the engint slghtly racing. It will not respond to the accelerator except to move at about 12 mph. Turning the car off and restarting it buys you normal performance for 30 to 45 minutes. The failures keep occuring at more frequent intervals until finally restarting does nothing. When my first battery failed I managed to drive 200 miles to get home. The next day I took to my friends garage and soon it would not move out of his service bay.
No way. A Prius does not run without the traction battery for several reasons. Firstly, there is no way how to start the ICE without the traction battery in the first place (ICE is started by MG1, which would not spin without a traction battery). And moreover, even if the ICE would run, the tranny cannot work without a battery pack; all currents are routed through it and it works like a "retention pool" of charge. So, to put it shortly, a Prius can go with a crappy traction battery, as long as it has enough power to start the ICE and enable the tranny to work (sort of), but it definitely won't run completely without it.
How much are they selling it for? Did you compare it with Blue book, Black book, and auction values? My '03 has 200,000 miles and it's running fine except it needs a new catalytic converter.