After the battery is disconnected or replaced, the engine stalls when it idles or is shifted into gear.
The left rear brake makes a "clicking" noise when it is used, and mechanics can't find anything wrong with it.
Sometimes when the gas is applied, the car jumps forward, and it sounds like the transmission is skipping, but it doesn't do it when we take it to the mechanic.
The car is very smooth, and is pretty quiet.
The car gets pretty affordable gas mileage.
"After the battery is disconnected or replaced, the engine stalls when it idles or is shifted into gear"
Quite common actually. That's the ECU attempting to relearn the functions of the sensor. It can sometimes take upwards of 90 days (!) for the ECU to fully relearn all functions, including the 100+ positions of the IAC. You can take it to a dealer or competent garage and they can retrain the computer, but doing that would cost some cash. Just ride it out if you can, and when the car stumbles, don't tap the gas.