2. If the light stays on when it is supposed to go out, they did NOT fix it.
It could be the case that the electronics that controls the "alert" lights has a problem. If this controller is part of the "master" computer, then that indicates that there are other problems in the circuitry. This is not good.
If the problem was part of a recall issued by the NHTSA, you can research it on the Internet.
At Cars.com, you can input the year, make, and model of your car to see what recalls are listed for your car.
Another strategy might be to contact the manufacturer's customer relations department and relate your experience. It may be that the dealer you took your car to is covering up incompetence, and the manufacturer may well take your side in the matter.