Paul,
I have the exact same issue with my late 2004 iBook G4 14". About 3 weeks ago it started shutting off for no reason. If I took the battery out completely, then put it back in, then do a PMU reset, it would fix it for a week or so. So I took it to the genius bar yesterday and they ran a diagnostic and it showed it was not a hardware related issue, then sent me on my way. Now some really weird things are happening as described by you. But now if I take my battery out completely, it will not even turn on. If you go to you system profiler and look at the power settings, this is what mine looks like, does yours look the same?
Battery Installed = Yes
First Low.. = No
Full Charge Capacity = 3898
Remaining Capacity(mAh) = 424
Amperage = 0
Voltage = 14294
Cycle count = 84
AC Adapter cannot charge battery = Yes
AC Charger Watts = 26
Connected = Yes
Charging = No
What does yours look like?
Tracy