Oil isn't the problem, my car will turn on again before too much oil drains out of the engine and oil pressure sinks too low. Cooling is, interstingly, so they switched to an electric water pump for cooling - it will stay running during auto off so you don't get too hot temperatures in the turbo for example. (4cyl 2l engine has a turbo)

There's quite a bit of tech going into auto off since it first appeared. (Heh, my first car in 1989 had an aftermarket start stop switch to stop the engine at the traffic light - i never used it except once or twice for fun - still one cylinder exploded one day on the Autobahn)