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Before you get too far into it and spend too much time or money - do a compression test on the engine.
Engine compression should be between 145 PSI (min) - 218 PSI (max( for each cylinder, with no more than 15PSI difference between the cylinders. If compression is low - shot a little oil into the cylinders and retest. If compression stays low - you have no choice but to rebuilt of do an engien swap. If compression improves, you need to do some diagnostic work to findout the real problem, most cases - it will eventually go to rebuild or swap.
If compression varies too much - no amount of fiddling with the bolt ons will help. Other things to test are injectors (there was a redesign in the later Corollas to improve durability and spray pattern), make sure there are no vacuum leaks, making sure the engine gets enough fuel. Should check for spark on each plug - possible there is poor chassis grounds or an excessive amount of engine noise that is messing up the ECM or signals running to the igniter/injectors.
All that P0420 tells you is that the downstream O2 sensor is not reading proper catalyst efficiency. This could mean the catalytic converter is bad - but not always. Could be a bad downstream O2 sensor (have to backprobe it while in operation to be sure) or the ECT sensor is faulty.
The 1ZZ-FE engine is pretty noisy - even when running normally. Ticking noises should not be an issue, unless they stay the same or get louder as the engine warms up. Need a bit more info on the car - like mileage, if you notices any oil consumtion, etc. - to get a better idea of what it going on and help point you in the right direction.
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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