What car do you have?
The ECU is "normally" rather durable unit, and if it fails the car is dead. Rather, inputs into the ECU cause most of the problems. Testing it is tricky and requires special tools, I would suggest getting a bentley shop manual for more info on this and the rest of these items.
As far as testing the coil goes, you'll need an ohm meter. But I would just check the spark quality with the plug. Also check the plug wires, cap and rotor.
Checking the injector function with a screwdriver works to an extent, sometimes the "clicking" sound can be faint. A mechanics stethoscope works better, but the Noid or test light method is best.
Now a "gap sensor?" I'm unsure of what you mean, do you know if it is called something else?
If your car is running rough at idle, I would suggest cleaning out your throttle body and idle air control motor, crud builds up in there and causes rough idle, stalling when cold, and surging when warm. Carb cleaner and a rag gets the job done.
Also check the cold start injector (if you have one) by unbolting it from the intake leaving the fuel line attached. Give the engine a quick crank and watch the injector, it should spray a very fine mist, if not, it can be cleaned (with some effort) or changed. But this wouldn't cause a cut out problem, just hard starting, or stalling when cold.
Your garage "should" be capable of testing and fixing all of these. If not you may need to find another garage.....