Newbee here, intro and pending questions

If you have those codes, they would make a great addition to a post asking for advice and support.

 

I'm guessing that last one was really P0420 mistranscribed as a U.

The P0420 and P1116 are two different issues, one with the catalytic converter, one with the coolant heat storage system.

The P0420 is detected by the ECM using a process that tries to learn how much "oxygen storage capacity" the catalytic converter has. Some time while you are driving, the ECM will pick a time to do a test. For several seconds, the ECM tweaks the fuel injection volume to make the mixture slightly wrong on the lean side, which leaves extra oxygen in the exhaust, and the catalyst is able to store some of that. Then, for several more seconds, the ECM will make the mixture slightly wrong on the rich side, but the downstream (of the catalyst) oxygen sensor will not show that right away, because the oxygen stored in the catalyst can make up the difference. Based on how long it takes for the downstream sensor to show a lack of oxygen, the ECM can calculate the capacity of the catalyst. It sets P0420 if that result comes out too low.

That process involves the catalyst itself, and both the upstream and downstream exhaust sensors and their wiring, and miscellaneous possibilities like leaky exhaust connections. If the catalytic converter was recently worked on, it's possible some detail there was overlooked. Or maybe the eBay converter wasn't up to snuff. There are about seven pages of troubleshooting steps in the repair manual (more info) that can help you pin down that cause.

The P1116 comes from a test of the coolant heat storage system, where the temperature sensor at the outlet of the thermos can be compared to the temperature sensor in the engine.

First, when the pump first starts moving hot coolant out of the thermos, the ECM looks at the outlet temperature sensor alone; it should show that the coolant coming out of the thermos is hot. If that reading doesn't change by at least a Celsius degree*, the ECM figures the sensor is dodgy or the coolant isn't really flowing for some reason, and sets the code.

After that, when enough coolant has been pumped between the thermos and the engine, the ECM compares the two temperature sensors' readings. They should be close. If they are more than 25 Celsius degrees apart, again, either there is a reading problem or the coolant isn't circulating correctly for some reason, and the code is set.

There are just a couple pages in the repair manual about troubleshooting that; again, it will come down to finding some reason for bad sensor readings, or some reason for bad coolant circulation. For checking out the sensors, these steps refer to the steps for P1115, giving you five more pages of ideas.

For communication codes (like U0100 and U0111), it can be helpful to have a scan tool that not only shows you what codes are there, but also shows you which ECU in the car was reporting each one. The reason is it takes two to communicate; when A is saying "I can't communicate with B", B is what the code tells you, and you get A from which ECU gave the code.

Again from the repair manual, it looks like there are two possibilities for who said U0100: the HV ECU and the battery ECU both communicate with the ECM and will say U0100 if they can't. U0111 is "hey I can't communicate with the battery ECU" and apparently only the HV ECU can say that.

The troubleshooting for communication codes will mostly be a matter of looking in the network diagram for how what is connected to what, and looking at the codes for what can't talk to what, and looking for problems in the locations fitting that story.

The key repair manual sections for that stuff are the "multiplex communication" and "CAN communication" sections.

* there seems to be a rare Toyota manual misprint here. In the DTC Detection Condition box, it says 3 ℃ (1.8 ℉), which ought to arouse suspicion because the conversion is wrong. But later on the same page it says 1 ℃ (1.8 ℉), which looks right.