What is limp home mode?
When the automatic transmission can't function properly, I believe it defaults to third gear to allow you to drive the car, but certainly without the advantages of using any of the other gears.
What kinds of questions should I be asking or having him check?
In addition to all the above, have him hook up his diagnostics device to check for any fault codes. Have him look at the oil on the dipstick, to see if it's really cruddy, and have him check the concentration of the coolant to see if that's where it should be (and may indicate that it's been properly maintained). Similarly check transmission fluid, to see if it has a burned smell to it (not good). Oh, steering rack, too, for any excess play. Shocks for signs of leakage. Condition of brakes and rotors. Having fun...? See if he can run an exhaust analysis to see if you're within spec to meet the next inspection.