Welcome to JustAnswer.com! My name is DriveFast71(Ron) and I will do my best to answer all of your questions completely and accurately.
The P0420 is a fairly straight forward code pointing to a fault with the Catalytic Converter. The Oxygen Sensors monitor the amount of emissions in the exhaust going into, and then out of the converter. When they see the converter is not 'cleaning" the exhaust enough to meet Toyota standards, the P0420 is set, and the converter is deemed "below threshold". Most commonly, this means the replacment of the converter. Best bet is to monitor the wave lengths of the oxygen sensors (this requires the use of a scan-tool) to make sure they are functioning properly. But, most commonly, this is straight forward, and the converter needs to be replaced.
When the ECT (engine coolant temp) sensor is faulty, it will set the appropriate code for this (P0125). The Toyota OBDII system is really a tight system, and when something is faulty, it pinpoints it, and sets the appropriate p-code.
When replacing the converter, you'll need to get as close to Toyota OE specs as possible. Those "universal" or "cheap" converters they sell at AutoZone will NOT solve the problem. I've had fairly good results with "Walker" and "Bosal" brands. Try to use a "better quality" parts store like NAPA. On this vehicle, the converter is built into the Front Pipe, and the Toyota OEM Part# XXXXX: 174100A310