Code 53 is the right O2 sensor and code 13 is the coolant temp. sensor. The bad o2 sensor will just cause it to run crappy, shouldn't cause any starting issues. The bad coolant temp sensor will also cause it to run crappy, but could possibly not let it start. What happens is the sensor is telling the ecu that the engine is either too cold or too hot (usually too cold) so it dumps fuel and can flood the engine. I had the same code and got a new sensor off ebay, codes gone now. This sensor being bad can also cause weird idle issues.
Even if you replace the coolant temp sensor, it still might not start. It sounds like you have some other issue that the ecu isn't catching. Could be a bad ignition switch, PTU, PTU harness, or some other electrical problem. If you get it running and it stays running it's probably not fuel pump issue.
You should check out this thread if you haven't already:
everything-you-need-to-know-about-the-z32-t508377.html