Firstly when you talk of mechanic only then it means that you can do denting/painting/accident assessment or that area is not of your concern tight now.
Mechanically there are few major items which should be checked since keeping a used car means that minor problems will come up every now and then. here is how to check major mechanical parts
Engine : blow is visible if u remove dipstick, a mechanic can easilt tell if a slightly thich oil is used to cover blow
suspension : a drive of 2 mins can easily give you complete idea of suspension (i'm considering that your mechanic knows his job)
gear box ; as above
ac : no extra-ordinary noise and some cooling
Now for other areas u didn't mention but which r also imp
-- frame and chasis be checked for accidents
-- chasis no and documents for fraud, tempering, etc
-- rust on body
-- touching / denting / painting : just be checked to assess the right value of car
-- any assembly with problem should be considered to be rectified in max possible expenditure e.g, if the seller says that ac is not working due gas, belt or switch, etc problem; you either count it as a burnt compressor or ask seller to get it functional before the final payment