The ground WIRE is the problem. The ground on the switch end is good, if it wasn’t the other side and your signals would not work as it switches them the same way. What you have is a bad connection between the switch and that light somewhere in the harness. If the circuit was good the light would work, you just need to find the break or disconnected wire.

Correct. The Brown wire should be hot when the lights are on. It is the power feed to the light. The Brown wire that has the white stripe is the ground wire and it is turned on/off by the turn signal switched. That is known as a ground side switched system and is very common on vehicles. Now you need to look at the Brown wire with the white stripe on it. In normal operation it isn’t connected to anything. When you turn the switch so the right hand turn signal comes on, then the wire should be grounded. As you have power on the other wire the problem is in the Brown wire with white stripe.
You cannot just ground the light so it stays on because that wouldn’t be the correct operation of that light.
You are doing fine so far, electrical diag claims a few brains every year… LOL
I don’t think it’s the socket. An easy way to test that, find the pin that the brown w/white wire goes into the plug, connect the test lights clamp to the battery positive post, now turn on the key and turn on the right hand turn signal. Now go to that plug and touch the connector pin that the brown w/white wire connect to. Does the light come on? If it does then it is in the lamp housing, but I’m betting it doesn’t.
You already know that the ground point is OK, because the turn signals and the other turn light work. So it has to be from the switch out to the light. Follow that brown w/white stripe wire from the plug back in the harness and I’m betting you find the problem

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The remote start shouldn’t have anything to do with the light. I remove things like that because many are installed the way yours is and they end up causing weird problems when a wire finally corrodes off or one of those twist joints comes apart. Especially when it is possible to install them much better for maybe 4 buck more in crimp and seal connectors and proper splicing with glue filled heat shrink.
Personally I would hunt down the manuals for both of those items then trace out the wiring and if you want to keep them, install them correctly. Or if you don’t you can see which wires do what and then repair the cars wiring back to stock. Take a look and see if these are correct -Avital 4200 (pdf) Bypass module pdf

No problem. A quick way to find a bad wire that is inside of a connector or even an open run is a simple tug test. A good wire can be pulled very hard with no issues, a bad connection will pull apart and if it was a corroded wire inside the insulation you will see it hourglass over the break before the insulation fails. If you want to see videos of this try this guide it might help. If you need more help you can ask there as well.

Light is generally better for this because it shows that the wire can carry a load. Think about a wire, made up of say 20 strands of copper. You can break or corrode all but one strand and it will still show 12 volts at the other end, but put any load on it and it stops working. No different than towing a load, a 3" rope or a piece of sewing thread will both connect them, but which one can take a load?
If you are measuring for voltage drop or resistance than the meter is good. For other testing I have everything from a common test light to a headlamp bulb that I use.

What they are doing is a top cylinder cleaning, basically the run the engine to heat it up, then you fog in a cleaner into the engine through a vacuum line enough that it shuts the engine off, then you let it set for a while, then start it back up and see if it helped free up the rings and carbon and did anything to help the engine. The thing is that there is no real way to know if it is going to help until you do it.

Hit an autozone or other parts store that does free scans, one of the things they can do is read the monitors.

^^^^^^^^^^^^^^^^^^^^^^^This is the correct answer…

That module does control all of that and is the likely suspect. Because of the design you can’t do anything like bypass it for testing. You basically have to check power/grounds and then look at the network signals to see if it is active or not and go on that. Sounds like that is what they did and determined the module is the problem, it’s actually somewhat common as a failure item.

I’m betting it’s a broken/cut wire or an unplugged connection from the fender bender. You say all the other lights work, including the right turn, if yes, it isn’t the switch, it triggers both the cornering and turn lamps. Simple way to test. Unplug the connector at the light assembly, Take a test light and connect it to ground, check the Brown wire for power. If it has power you are 1/2 way there. Now connect the test light to battery power and start tracing the Brown w/white stripe wire back to the switch. With the turn signal switch turned on for a right turn (key can be off) the switch completes a path to ground. So if you test at the first connector (C1444) and find nothing, go to the next one (C134) then on to C260. I’m betting you find it out in the harness near the other damage.

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Depends on the state. If you are in NY or CA the cat needs to be CARB certified and have an ID number that matches the CARB chart for that engine family and car fitment.