Coolant problem, I have been having to add coolant every 2-3 days?

If the wires for that valve are green and white, and there's only 2, it's the Purge valve and should be closed when it's not active.
As for the 2nd picture, it depends on where those hoses go, it does look like something is missing there, at this point who knows what they have taken off or broken, but that valve you have off looks like the purge valve to me with those wire colors, the MAP sensor is 4 wires because it has a temperature sensor built into it as well. I'll post the wiring diagrams so you can check it.

1st diagram is the Purge valve which pulls fuel vapor from the gas tank, should be closed when off. The 2nd diagram is the MAP/IAT sensor. With the 3rd the wiring going to the ECM. Hopefully they didn't remove anything, in the picture it almost looks like something is bypassed on that hose, but I can't really tell.

Don't touch this!

Yeah 02 sensors can be difficult to get out, I was surprised the impact gun didn't strip it out, Try getting it out when the exhaust is hot, just make sure to wear gloves so you don't get burned. They tend to come out easier when the pipe around them is hot and expanded some. The MAP sensor should be on the back of the intake manifold, the MAF (Mass Air Flow) sensor is the one that you would find on the air intake tube. It would be before the throttle plates, where a MAP sensor will be found after the throttle plates. (Its Manifold Absolute Pressure) so it gives a reading of intake manifold pressure compared to atmospheric pressure. Hence the "Absolute" part.
So, there's actual carbon build up inside the Purge valve? When it comes to wiring diagrams, I post them mostly so people can compare the wire coloring to the component they are looking at. For the Purge valve I could see a 2-wire component with one green wire and one white wire. So, you can do the same for the MAP sensor, although the location I see is not the one in the picture, where something appears to be missing.
This vehicle has a MAP and a MAF sensor.
If you can't get the front Oxygen sensor out for a back pressure test, if you're worried about the Cat, you can put a vacuum gauge on the intake manifold and it should read around 18- 20in of vacuum at idle, roughly. So, if the cat is clogged (or restricted exhaust) vacuum will lower.
But a back pressure test at the front oxygen sensor location at idle should not be more than 1-1.5psi max. If it's higher or jumps way up on a throttle snap, you'll know the cat is having issues, I will usually back up a pressure test with a bore scope down the exhaust to visually see the condition of cat.

These are a couple of pictures I took of a vehicle that was setting a P0420, Bank 1 Cat efficiency code, even though you can see through this cat, it wasn't storing oxygen like it should and therefore was not able to burn off the combustion gases. I was surprised by this one, I expected to see it melted down more, but when comparing it to the new one, there was a definite difference. This was caused by a bad front Oxygen sensor (A/F-air/fuel ratio sensor) on this V6 Nissan. I think it would have probably passed a back pressure test, but this was verified by monitoring the rear Oxygen sensor voltage, it was fluctuating like I described in my other post, the voltage wasn't steady at all. And it would set a P0420 code pretty much on the first drive after erasing the codes.

What is the Bank 1 Sensor 2 Oxygen sensor voltage doing when the engine is up to operating temp and just cruising under part throttle?
Thats what you really need to see, of course, though you need to be running on the engine and not electric motor. So, under throttle load it should switch over.

Is your scan tool able to record live engine data when driving? Thats the safer way to do things or have someone else drive so you can observe the live data. With a hybrid setup the difficult part is knowing if the gas engine or electric motor is doing the driving. So, diagnosis won't be the same as a normal combustion engine, I see there are some lamps to tell which mode the vehicle is in. EV mode for electric motor only and if that lamp is off, the engine is running.
But as for the rear oxygen sensor try to keep the throttle pretty steady when monitoring the voltage levels, remember when you let off the gas pedal the ECM will shut down the fuel injectors during fuel cut-off, its does that when de-accelerating because the engine doesn't need fuel at that short period of time, so the oxygen sensor is going to read lean during fuel cut-off. Such as coming off a highway and coasting to an idle, the injectors will be turned completely off at that point, it's to help with emissions.
I wouldn't mess with the throttle plates, some of them are not supposed to be moved by hand, it can damage the motor inside the throttle body. I would concentrate on the codes that are setting right now,
Do either of those codes come back right away if they are cleared out? I would suspect one of those settings during the EGR cooler leaking. And possibly not cleared out after the repair. Are they reading as "Current" codes or "History"? And intake manifold leak can cause the EGR code, if theres any hoses not properly sealed and manifold pressure is incorrect during the EGR test period its going to fail that test. I've been trying to find a decent picture of all the hose routing for the valve cover and intake, but many of them are drawing type diagrams, it's tough identifying that hose in your 2nd picture.

Oh goodness, for checking the Cat condition via the Oxygen sensor voltage you'll want to be monitoring the rear oxygen sensor, B1S2 for any voltage fluctuations. Its the one that should remain pretty steady, the Front oxygen sensor (B1S1) is an A/F sensor (air/fuel ratio sensor). It is a much more advanced wide band type sensor. Thats why it has 6 wires, while the rear oxygen sensor only has 4.
Since youre getting oxygen sensor heater circuit faults now, Im not sure why youre ending up with so many issues like this, but both B1S1 and B1S2 heaters are both fed by the same fuse. "SENSOR 1 Fuse 15amp". Both sensors are controlled by the PCM on the ground side of the heater circuit by pulsing the circuit on and off (PWM).
If the heaters are not working the sensors will not respond correctly and these codes will set. I wouldnt bother putting any more parts on this car until at least the oxygen sensors are not setting anymore codes.

But if the vehicle is leaving the shop with more issues than it went in with, they are responsible for those issues, you're running into too much here to diagnose in a logical manner. Especially with how many times electrical connectors have been apart. Every time theyre pulled apart and jammed backed together, now you have to worry about pin tension in each of those connectors, if you think about that, you could potentially be chasing a number of wiring issues. You're going to become overwhelmed by all this before you know it. It needs to go back and they need to figure out some kind of deal with you,
but check that fuse and see if it's blown.

Well, some codes are Pending some are Current or Permanent, so the PCM is seeing faults under different conditions. The PCM is monitoring intake manifold pressure when opening or closing the EGR valve, so its doesn't have to be an issue with the valve itself, there isn't that many components failing at once. Thats not possible. This is an electronic EGR valve, it does not have a control solenoid, it has an electronic motor and a position sensor that acts just like a throttle position sensor, The movement of the valve is 0v-5v.

The solenoid you pictured was the Purge valve, two wire green and white. You can't really mess up the oxygen sensor because the front one is 6 wire and the rear one (after the cat) is 4 wire. They are completely different types of sensors, the front is a wide band for more accurate air/fuel monitoring, the rear is a narrow band oxygen sensor.

I think you are dealing with a bunch of wiring problems, heat and vibration, driving time, etc. Contribute to when and how much electrical connectors are heating up and cooling down, pins expanding and contracting, that's why codes are coming and going, the dealership has just been at this wiring harness too much, they replaced the engine over a coolant leak, that should tell you something.
Instead of spending more money on parts, spend that money on a decent shop in your area, that has good reviews and they can at least give you an honest idea of what to focus on right now. It's impossible to chase so many problems at once.
Let someone get a fresh set of eyes on it, trust me it will be a relief for you when they come up with just a couple or one area to focus on. Just ask if they have someone who can work on hybrids.

No, that is all incorrect. The oxygen sensors are strictly input devices, they do not control voltage to anything. The sensors only measure the amount or lack of oxygen in the exhaust, thats it. The EGR valve has no vacuum control at all, I showed you the diagram, it is an electric motor that opens and closed the valve. There is nothing to do with vacuum on modern vehicles. Dont replace anymore sensors until you find out if the oxygen sensor heaters are working. Neither sensor will function without their heaters. Ive posted service info on the operations of these systems, they do not function like older vacuum controlled designs. All that is gone. Dont spray any cleaner into the Purge valve, youre going to ruin the valve. You're disregarding all of the information I have posted so far, when a code sets it does not mean that component is the cause, its the result of incorrect operation of the component, therefore you need to follow service information on how to correctly test the component and its control system.

This is all that controls the EGR, two wires, thats it. You cant guess at how a circuit or component works, youre all over the map here, you cant fix something if you do not know how it works. Did you check the fuse for the oxygen sensor heaters with a test light yet?
Here is all the pin point testing for the EGR valve.

Don't put a cheap upstream sensor in (B1S1). Only use an oem. These A/F ratio sensors are wide band sensors that are extremely accurate in their measurements. They work off of current flow instead of voltage like the rear 02 does. The EGR valve assembly does not work off of vacuum in any manner at all. Im not sure why you keep mentioning vacuum when it comes to the EGR valve. The EGR system only opens and closes a small electric motor to allow a small amount of exhaust gases into the intake manifold under heavier engine loads. The added exhaust gases are introduced to strictly lower combustion chamber temperatures and this prevents NOX gases from being produced. NOX is a harmful emissions gas which, by law, needs to be extremely low. Thats why the EGR tubing goes to the intake manifold. The code is most likely setting because there is a leak somewhere thats skewing the test results. If theres an intake manifold leak, and the PCM opens the EGR valve during its self test, the PCM will not see enough pressure change during the EGR opening because the intake leak has already lowered pressures too much, the PCM will then conclude there is insufficient flow, because the pressure is already too low to begin with.

Pull up your live engine data and look at the Long and Short Term Fuel Trims at idle and 2500rpm, but try to check this when the combustion engine is running, not the electric motor. If there is an issue with the front AF sensor giving incorrect data the Fuel Trims will be off. Fuel Trim is the air/fuel ratio compensation.
Fuel Trims will also be effected if there is an intake leak. This is probably the cause of the p0139 code as well, unless the oxygen sensor heaters are not working. If the heaters are not working, it doesnt matter how many times you change the sensors, if theres no power to the heater circuits the sensors will just sit there and do nothing. >>> Fuse blown?

That white carbon is due to that coolant leak. The Purge valve with the green and white (2 wires) should be normally closed when it's unplugged. So if you can blow through the old one, then its no good. Your scan tool data should list the EGR position in engine live data, and if you pull heavy throttle you should see the EGR position opening, Its used under heavy load/ heavy throttle because thats when NOX gases are produced. I thought they had replaced the EGR valve already?