The red lights stay on for 30 seconds, I would guess.
Very occasionally, and I'm talking 1 in a hundred times, it will remember the position.
Are you using an "official" cable or a scratch built one?
I'm pretty sure that the Phatbox saves its current position in memory and not on the DMS. I'm basing this on it going back to the first playlist after disconnecting and reconnecting the power on my BMW Phatbox. I can't see why the other firmware versions would be different.
The Phatbox uses two different sources of 12V - one that's always on, and one that switches on and off with the car (possibly with delayed power-off as the car does a shutdown). Note that some car models may not use separate wires for these - for example, the BMW changer interface has only constant power, ground, and a bidirectional data line. The Phatnoise BMW adapter and cable route the constant power to both the constand and switched power pins of the Phatbox.
It may be possible that the switched power is turning off before the Phatbox has completely processed the "go to sleep" command issued by the head unit.
What happens if you either select a different source (for example, radio) on your head unit or turn the head unit off a few minutes before shutting the car off? On the BMW version (which is the one I'm most familiar with), selecting a different source sends a stop command to the Phatbox. Turning the head unit off sends a stop command followed by a shutdown command to the Phatbox (it takes a minute or so). There has to be some command like this in all the different firmware versions, as otherwise the Phatbox would keep playing (silently) if you switched to the radio and back to the Phatbox, which would mean you'd be somewhere else in the song or even on a different track / playlist.
If you have a startup sound configured on the Phatbox, you can tell if it was in stop mode or completely shut down - if it was shut down, it'll play the startup sound when you power on / select it on the head unit. If it was just stopped, it won't play the startup sound.