Its also possible the alternator got damaged in the backwards jump start incident. A diode might be shorted out, which would explain the battery going dead with the engine off. And it could also explain the battery getting over-charged when the engine was running. If you can get it to thepoint it will start and idle, first check is this: With engine off, before first start of the day, battery should measure about 12.6 volts. Immediately after starting the engine, 13.5-15.5 volts.
Ok thank you will check that
A failing alternator won’t prevent the engine from starting as long as the battery has sufficient charge. Connect a scan tool and see if the PCM is alive, it sounds like the PCM has failed.