Yes, the computer should operate fine without a battery. In fact, for testing purposes, it's ideal to have the battery removed, in order to rule the battery out as an issue.
The fact that the power plug is green is a good sign, but it does not necessarily mean that everything is functional. Does the light flicker if you twist the plug in its socket? This could be a sign of a bad AC adapter, or a bad DC-in (the plug in the computer).
PRAM is irrelevant because the computer must be starting up in order to do a PRAM reset, but definitely try testing the PMU (hold in the power button for about 10 seconds while connected to AC power). Next I'd try a different AC adapter if possible. I would also remove extended RAM and Airport, and try the resets again. If you have two RAM slots and both are occupied, try powering on with a memory module in one slot at a time in order to test for a bad slot, which is common with PowerBooks, and typically causes them not to boot, just as you are experiencing.
Also, if you have a 15" or 17" Aluminum PowerBook (silver keys, as opposed to black), you need to be using a 65W AC adapter, and your machine will not respond with a 45W AC adapter meant for less power-consuming machines. If by chance you're using a different AC than usual, and it is 45W, then that could be the issue. If it's an Apple AC it will state the wattage on the side.
Good luck, and let us know what happens.