If the catalytic converter is broken inside it can cause the car to run rough and even shut down on you. This usually happens when the engine is up to operating temp however, it's less common on first start up or when the engine is cool.
I would have the computer scanned once again for codes because it can store more then just one. The symptoms sound more like an issue with a cam or crank position sensor. If the computer loses signal from these it will shut the engine off. Sometimes when that happens, it takes only a couple of minutes and it will start right back up for you.
The jumping could be consistent with the converter blocking off the exhaust however. I think it would be important to lower the exhaust pipe and check and make sure it's not broken up inside.
Fuel pump is possible, but usually when they fail, they fail and that's it. They don't tend to fail then suddenly come back to life. A fuel pressure test could at least determine if it's weak. If the pressure is low then it should be replaced.
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