My 2003 A-4 check engine went on and the MPH1 light along with it. It is running good otherwise.

Hey Connie, There are numerous reasons the check engine light will come on... from a loose or not sealing gas cap, to numerous sensors...if it runs great, gets good mileage..>>>???

If it won't start now, you would need to check for spark to the plugs and fuel pressure along with trouble codes in the computer.
You'll need to know what direction to go in to proceed.
It's possible the computer shut down the engine if it was too hot, but your temp gauge should have reflected that.

Have the engine running and wiggle the ignition key,seems like you got a loose connection in there.

Engine light flashing along with the drop on RPMs is indicative of misfires. Connect a scan tool and see if any codes are stored this may give you a starting point. As for the clicking I'm not sure that it is related to the engine light but clicking when braking may be due to worn suspension/steering parts such as control arm bushings

need to have vehicle scanned the trouble codes will be stored in the computer most auto stores offer thi service for free check ur local area there are a few hundred different codes or problems that can cause ur problems and being interment can also be an electrical connection/harness problem

As long as it does'nt stay on it's doing what it's suposed to be doing which is saving you from going out of control. It'll come on more often if you drive like hell, run with low tracktion tires or worn out ones and if the air pressure is low. If your tires have over 80,000 miles on them even if the tread is still good their bad tires as their skating along and the trackstion system is working hard to keep the car under control. If the tires aren't al the same size it'll do it to. ( fronts worn out backs still with tread.) Mis matched tires too. Ifthe light stays on other light's are bound to be coming on too the ABS light the brake light, and even the engine light as their all involved in traction control.

first check ur gas cap if wasnt put on correctly ,left off ,or does not seal usually sets off one of the evap codes--if cap looks good than need to check vent valve possible faulty and the wiring circuit for damage or short -also check vent valve for blockage

Some auto parts stores (like AutoZone) will diagnose the code for the check engine light for free. If the vehicle is running fine it may be something as simple as a bad oxygen sensor.

check your engine and battery ground wires and the fuze links.