What is your opinion on this noise, shake and missing?

This is not something that can be determined over a computer. If that were possible, we wouldn't need mechanics to diagnose problems. They could just punch in the symptoms, wait for a reply, then replace parts.

I CAN offer a few suggestions though. First of all, when you said, "Every single solution for this code has been done", the vehicle didn't come from the factory doing this, so you know something is different. I have a suspicion the mechanic has fallen victim to a very common problem. He is likely simply going by the diagnostic fault code, then replacing parts. In fact, diagnostic fault codes never say to replace parts or that they're defective. They only indicate the circuit or system that needs further diagnosis, or the unacceptable operating condition. When a part, most often a sensor, is referenced in a fault code, it is actually the cause of that code about half of the time. Reading the codes just takes a few minutes. The diagnostic time you're paying for is for the mechanic to do the pages of tests for that circuit to rule out everything other than the part. When wiring and connector terminal problems have been eliminated, and only the part is left, that's when it is proper to install a new one.

The next thing, even if you're getting a variety of fault codes, we need to know the exact code numbers. Some parts, like oxygen sensors, can have over two dozen potential codes, and they all mean very different things, and require different diagnostic steps. In the case of the EGR system, there is a circuit that tells the Engine Computer when that valve should be opened, there's a circuit that opens the valve, there's a circuit that tells the computer the valve opened, and there's a circuit that lets the computer know exhaust gas is flowing. Insufficient flow can be caused by the valve failing to open, by it failing to be told it's time to open, it may have opened but that wasn't reported back to the computer, or the valve opened, the computer has verified that, but it isn't seeing any flow. You could simply have a tube that's plugged with carbon.

For a clicking noise, you have to have a live person to listen to it and determine where it's coming from. Common causes when it occurs with the vehicle standing still include an exhaust leak near the exhaust manifold, lifters or lash adjusters not pumping up with oil, the serpentine belt coming apart, worn bearings in the generator allowing the rotating field winding to hit the stationary winding, a loose timing chain slapping against the cover, even an emissions-related valve cycling on and off. Those valves typically click twice per second and start being audible after a foam silencer pad falls off.

Most importantly, when you have an engine-related issue, you need to list the engine size. You also didn't list the mileage. We use those details to come up with replies of value. There are some things that are real common on a 20-year-old car that would never pop up on a two-year-old car, or on a low-mileage engine. Given the additional symptom of a shaking engine, that is due to a misfire which any competent engine performance mechanic should be able to identify. If the spark is jumping somewhere where it bypasses the spark plug, you'll have a misfire and you'll hear that as a snapping noise that changes speed with changes in engine speed.

Compression is a factor of how well the piston rings and valves seal. It will go down a little with high mileage, but it will not be intermittent. If a set of symptoms goes away at times, compression has to be okay. Testing it will be a waste of time.