P0420 is the Catalyst System Efficiency Below Threshold.
For some reason the post and pre Cat O2 sensors are reporting similar or one is reporting an unbelievable signal.
Due to the rough running, I don't think the Catalyst is the problem, but something effecting the readings of one or both of these sensors.
I'd guess that for some reason the car might be over fueling (tripped into or fooled into it) and the Cat can't cope with the results of the extra rich fueling.
It's fairly common for O2's to fail, one way or another.
They have two parts to them.
A heater element, so they self heat to work quicker on a cold engine
And
The sensor part it's self, though not really a sensor as such, but a electric signal generator (heat from the exhaust is turned into an electical signal, the hotter the gas = the leaner the engine thus a higher signal is generated and vice versa)
I know from experiance the ECU on one of these has trouble detecting a heater element fault (not heating quick enough rather than open circuit) and can cause similar issues.
Obviously something may well effect these signals and often either sensor may be condemned by mistake.
Air leaks, exhaust leaks, coolant temp sensor issues, misfires and wiring damage to the sensors can all have an effect of the temp of the exhaust gas or what the sensor reads, so they need to be ruled out first.
Unless you are handy logging O2 signals and understanding how the modern fuel systems works and what is should be doing at various points, it can be hard to detect what the actual fault is.
Particularly as it's a self governing "closed" system, it can throw you right off the scent, for example a small air leak can in fact cause an over fueling issue, as the ECU adds fuel to trim the lean signal out until it can't anymore and flags a rich related DTC!
