P0420 Catalyst System Efficiency Below Threshold (Bank 1) could be caused by your fuel problem, if the system is thinking it's too lean and has been trying to adjust to that , but there are a number of other possibilities as well:
A damaged or failed oxygen / O2 sensor
Downstream oxygen sensor (HO2S) wiring damaged or connected improperly The engine coolant temperature sensor is not working properly
Damaged or leaking exhaust manifold / catalytic converter / muffler / exhaust pipe
Failed or underperforming catalytic converter (likely)
Retarded spark timing
The oxygen sensors in front and behind the converter are reporting too similar of readings
Leaking fuel injector or high fuel pressure
Cylinder misfire
Oil contamination
My guess is that your fuel delivery issue is creating other problems as the computer tries to compensate. In the process, it may have damaged the O2 sensor, but I doubt it killed the cat, since you say you were only able to drive it twice....unless it had a problem prior to you getting it. If it was misfiring, you'd be seeing the CEL flashing when you did have it running.
Check for wiring left unplugged, too. If they left the breather hose off, no telling if something else got left off, or not securely connected.
The P0480 Cooling Fan Relay 1 Control Circuit is also a little generic, and could be due to a number of things:
Faulty fan control relay 1
Fan control relay harness is open or shorted
Circuit electrical connections poor
Faulty cooling fan 1
Faulty coolant temperature sensor
Fan cooling fan harness is open or shorted
Cooling fan circuit poor electrical connection
Intake air temperature (IAT) failure
A/C selector switch
A/C refrigerant pressure sensor
Vehicle speed sensor (VSS)
Since you mentioned the mechanic thinking a relay was "not in the right place", I'm rather wondering just what he fixed, and what he may have disturbed in the process. Check the fan connection, and look for possible chafed wiring. Could just be that something didn't get reattached properly.
I don't think these codes are the cause of your no-start situation, but they may have cropped up because of the problem and subsequent efforts to fix it.
Good Luck!
Chris