Does the Raspberry PI support WOL (Wake-on-LAN)?

No, it doesn't. But the Raspberry Pi draws about 2 watts at idle and with a little effort can be reduced even lower than that. So there's really not much point. As long as you run your car every ten days, you can keep the Pi on all the time with no issues.

It might be worth the effort to rig a device to shut the Pi off if it remains on for a week without the car starting. (Just give the Pi the ability to shut off its power.) That way you can leave the car for several weeks without having to worry about the battery draining.

I do understand that full automation of this would be simply sexy and a make of it a "better toy" but I think that a simple and old button with a "on" led would perfectly do the job and the little pi would take a very little time to boot from scratch meanwhile you put your seatbelt on. And never underestimate the good old retro "Knight Rider" feeling :)