Multiple codes P0003, P0014, P0036, P0203, P0704, P2003, P0420, P0300, 301 and 302

Hello, this looks like a difficult situation. I would almost say this looks like either an ECM failure or a wiring harness with a bulk connector that has a serious issue. Do these codes all come back right away if you erase them, if so, do they come back with just the key in the "On" position, engine not running, or do you need to take it for a ride? Also do you have a own scan tool that can read codes and look at live data?
This could also be a CAN bus network fault. There are too many codes here for all those components to have failed at once. What have you replaced already?
Some of these codes are not even listed in the OBD2 lists. I'm not how some of these codes are being generated. There is no P2003 and the P0704 doesn't seem to apply to this vehicle being that this is an automatic transmission.

https://www.2carpros.com/articles/can-scan-controller-area-network-easy

Okay, and what sub-model is this? GLS, GS, SPORT? The GLS is the only sub-model that is a Sedan, the other two are Hatchbacks, but they all have the same 1.6L engine (G4FD) with a GDI High pressure Fuel system. I just want to make sure we are on the exact model for wiring diagrams. I don't see many changes between the three, but I would like to be sure.
So, since you have replaced the cam sensor, this code is for the Cam Sensor B, so it's the Cam sensor for the exhaust camshaft. Let's check that sensor to make sure it has its Reference voltage and a good Ground. As a starting place until you get your new scan tool.

The second diagram is the exhaust camshaft sensor plug, but go by the first diagram,
You can check this with it unplugged, key on, engine off. Since we're just checking circuit integrity. So, with a multimeter set on voltage and hook the black lead of the meter to Battery Negative. The grey wire should have five volts, The White wire should be as close to 0 volts as possible. The Green wire is the Signal back to the ECM, but without a lab scope it will be difficult to look at a running signal. But as you will see on the fourth diagram, it should read five volts. These are Hall Effect sensors, so it will generate a 0-to-5-volt square wave signal.
If things check out okay with the sensor, for the time being we are going to assume that these codes are being set due to either a bad ECM, a software update needing to be done. Or a communication network issue. There's no way to have that many failures, but you can also check the CAN bus network with a multimeter to see if there is at least a voltage signal on the network. One of these issues is also why you probably cannot erase the codes.
The engine computer (ECM) is right behind the battery on this car,

Also, I just came across a Technical Service Bulletin on an ECM upgrade. Although this TSB is for an issue with a vehicle with the Check Engine light on and no Codes stored. I believe this may be related to what's happening with your vehicle. These might be falsely generated codes where there is actually no problem with all these sensors. Some of the codes you're getting don't even apply to this car. The P0704 is for a Clutch Switch which this car does not have, also the P2003 looks to me like a diesel engine code.
This ECM is probably used in many other models. The ECM upgrade is for the ROM and is also saying during the update to verify the correct part number. It almost looks like the ECM may lose its memory or it wasn't retained.
The scan tool you are using is reading generic obd2 data, it's not reading the manufacturers data inside the ECM. That can also be another reason you can't erase these codes. I would check on any ECM/PCM updates or upgrades from the dealership first before you go putting anymore parts or computers into this vehicle. If a major upgrade/update like what I'm reading here is necessary, they will have the manufacturers scan tool and software to do it.

This looks more like a computer problem. Thats why after replacing all those parts the codes remain. If there was an actual fault with the cam sensor for example, and it was replaced, the code would become a history code because the ECM would be getting the correct signal from the cam sensor, so it would have no reason to keep the code in memory. Same with all the misfire codes, there's a code for the number #3 fuel injector control circuit but no misfire code for that cylinder. But misfire codes for all other cylinders. That makes no sense. This points to a computer problem. Pages 5 and 6 below are the TSB for the ECM update.
Page 7 is another ECM/ECU update TSB from a different information source.

Well, that's very strange. I do hope it does too. It might have been in Limp Mode because of all the codes. But if that happens again, I would say the ECM needs an update.