Pandora doesn't have this. It's one feature that is really lacking on the GP2X - due to Linux no doubt.
Yes it does, and does it in the exact same way that Palm (actually, in a much better way than Palm) did. It has nothing to do with Linux or not. When you power off a Palm, it suspends to RAM: shut down everything except the RAM, so it maximizes battery life. The Pandora already has this. In the event the battery goes dead, however, you see a several second boot time on the Palm, just as you would on the Pandora, or any other device. What the Pandora also has that the Palm (at least the older ones that I'm familiar with. They may have changed it) does not is suspend to disk: take what's in RAM and save it to permanent storage and then shut off the RAM. That way, even if the battery goes dead, you just charge it up and you get your instant on again.