Anyone who relies solely on Carfax may wind up being surprised with some bad news later, as shown by a number of posts on this board. I think that your mechanic knows what to check, but if you want my opinion of what should be checked if oil change intervals are suspect, I would suggest taking off a valve cover and checking for sludge. If evidence of sludge is found, then I would suggest a compression check.
Overall, wouldn’t it be easier to limit your potential used car purchases to those that come with maintenance records? There are a whole lot of used Corollas out there, and surely I can’t be the only car owner who keeps all of his maintenance records.
I know that I am a bit obesessive on this topic, but I even maintain a small chart detailing the date and the odometer mileage of all services performed on my cars. This way, I can see at a glance when the serpentine belt was last changed, or when the brake pads were replaced, or when the transmission fluid was changed or… That sure beats having to go through a pile of invoices each time that you are wondering when a certain procedure was last done.
P.S. Incidentally, I would also advise you to have your mechanic check for possible collision damage, as Carfax DOES NOT always reveal an accident in the car’s past.