There's an oil pump on the EA888. No idea about the lesser engines:P
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most new cars come with water cooled turbos anyway so no need to cool them down, I don't bother with my Octy and I never did with my GTR either and it's turbos never had any issues and it was running 1 bar boost and 240AWKW.
If you're that worried just let it idle for 30 seconds before you turn it off. Don't bother trying to drive off boostas the turbos are so small they spool up too low in the rev range so you are always going to be making boost if you re driving.