Will a car run with a bad ECU?

Don't listen to the "Mechanic". While he makes valid points about the abilities of the alternator and battery, he is wrong about the ECU. The ECU does monitor the battery voltage, but one of its jobs is to send a signal to the alternator to induce a higher charge on the battery during higher than normal load conditions.

Now, if your battery will not hold a charge (assuming this is a relatively new battery), there could be a couple reasons. First check if you have a good ground and ground wire (-) from the battery to the chassis. Make sure your battery has tight connections. Also cut back the wire going from the positive (+) side on your battery going to your alternator. If this wire is corroded (yes, this happens more often than you think), then replace it. Also, check your fuses. On most models there is an "alternator" fuse under your dash or under your hood fuses. If all else fails, you would need to check the wire coming from the ECU side wiring harness and verify if there is output (this is usually last case scenario, and the most difficult to verify, so save it for last).

alternator can be good but dying battery is probably just too old try replacing this this doesnt sound like an ecu problem because the ecu only monitors the charging system the alternator is a mechjanically driven electric motor which charges battery and then battery charges it it is a back and forth system if one is bad it can cause many of issues you are experiencing

i would get that repaired or replaced, because the last thing you want is to have a blown ECU in the middle lane on the highway. or have the ECU start malfunctioning...

nope, with a bad ecu your coils will not send a spark to the plugs. have you changed your battery?

you mean the computer? yes it will run bad with a bad computer.