Why does my car's ECU have Alzheimer's?

The check-engine light has come on recently due to a different (presumably unrelated)
problem with the engine, so I take it to a mechanic. Mechanic calls me up and tells me
there are a few problems -- the first is the reason is the misfiring engine, and then he also
notes an error reported from the check-engine readout about my oxygen sensors. Which
leads him to a third, more sinister problem: Every time he turns the car off, the ECU or
whatever is responsible for keeping track of system problems (and signaling them in the
form of a check-engine code readout) keeps rebooting every time the car turns off.

That's not good. I'd like to know if the other issues with my car are occurring together --
maybe they are symptoms of a larger problem, for instance. The computer is supposed to
record and monitor whatever problems are tripping the check-engine light, but I notice that
every time I turn on the (still not yet fixed) car, the check engine light is OFF. Then it's
like the car suddenly remembers it has a problem, and only after about five or ten
seconds of idling does the dashboard signal a problem.