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

That's a lot of money, it's just Al btw. The site I saw it on is a super cheap kind of knock off type site, I've purchased some electronic parts from them before (non-automotive related) microchips and such, so I'm not sure about their parts, I was more looking for a part number, it was just aliexpress com. I think it's stuff directly from other countries. You are correct on its operation, the exhaust gases simply run through the center pipe with a cooling jack on the outside, you can see the 2 cooling pipes that have rubber cooling hoses clamped onto it, that's where the coolant circulates threw and around the center pipe cooling the exhaust some. I have the same assumption, that it cracked need one of the welds at that top flange, or just plain failed somewhere, but either way. To be absolutely sure, I would pressure test the coolant system with the bolts for the cooler on both ends taken out. You could buy a cooling pressure kit at Harbor Freight, they sell a cheaper one. But you might find it leaks when you take off the bottom pipe to the manifold. It has to be leaking quite a bit to go through so much coolant. There is a gasket on the top and bottom, you will have to do an online search for those. I don't see any in stock either. You have everything correct though,
This is the pressure test kit harbor freight has, you could use it once and return it I guess, unless you plan on working on your own vehicles from here on out.

Customs tax? That doesn't sound right to me, regardless of where you order it online. You might want to try some used parts lots, maybe find a crashed vehicle. I will look around in my area as well for you, I'm in NC, if I can find one, I will let you know

Wow. You can physically see the EGR cooler piping though, that was in the diagrams we were looking at, correct? It all looks the same as what is on your vehicle. I just want to be sure, it's sad when they bring up doubts like this. And I can't be at the vehicle to help you with this, that makes it even more difficult.

Maybe they don't know what it is, it's possible their techs are not up to date, I mean you and I found it by going through service info, but really, it's been there so many times, I don't know what the excuse is. I'm sorry it is taking so long to just get your vehicle back on the road running correctly. Let me know what eventually happens.

Wow, that's a lot to go through. Not a dealership I'd want to deal with. It's actually a really stupid argument on their part, they could have had a happy customer with just telling you the truth. Since it probably didn't need an intake anyway, but hopefully now this will finally be the end of all the headaches. Thanks for the update. I was wondering there for a while what the outcome was.

Okay, with the P0401 this can be more than just an EGR valve flow issue, I'll post the setting criteria and some possible causing for the code setting. The fact that it set after the repair, and wasn't setting before, tells me it was most likely something caused during their "repair". Such as moving the wiring harness around too much and stressing it, or not getting everything sealed back up completely like it should be. It's possible they didn't get the EGR tube completely sealed, or some other issue along those lines. Service info backs this up with possible "Poor or damaged harness connections" EGR pipe leaking" Intake manifold air leak" etc.
Diagrams 1, 2 are for P0401 and 3, 4 are for P0139, Id say the same thing for the P0139, those codes were not setting before. There should be some water dripping out the tail pipe under certain conditions, that's the Cat converting exhaust gases over to H20, although, with the amount of coolant you were losing, that is Cat contamination, I will dig into these codes some more later this evening. But I'd say once you get this thing fixed, you may want to think about getting rid of it, it's been nothing but headaches for you all the way here.

If its a hose coming out of the engine block, its most likely going to be some kind of PCV hoses. On newer vehicles they direct positive crankcase pressures to all kinds of places, but its important that its not blocked off and is routed to the correct intake position, that positive pressure needs some place to go, or it will start pushing out any seals in the lower end of the engine block, such as the front or rear main seals. Thats how those seals prematurely start leaking.
These are the PCV diagrams and really the only vacuum hose diagrams I see in Service info. Sounds like they might be a bit hacky when it comes to repairs at that location.
If you can pull up live engine data, and monitor the rear Oxygen sensor voltage, or graph its voltage is better, but get the vehicle up to speed, such as getting on the highway for a couple of minutes, so the engine gets up to temperature, before you exit, give it heavy throttle so youre not running on the electric motor, this way you should be pulling on the combustion engine, and when you let off the gas pedal suddenly, the fuel injectors should go to Fuel-Cut mode and actually shut off until you slow down enough where the engine needs to return to an idle, during de-acceleration fuel cut off, the rear Oxygen sensor voltage should go full lean, meaning almost 0volts. If you have the sensor data graphed you should see the rear 02 take a dive from its steady 600-700mv to less than 200mv (or 0.2v),
my truck goes all the way to 0volts on decel fuel cut off. Its a way to check the rear 02's operations, and its the way the ECM is doing it as well, but again you are correct about a possible Cat issue. (6th diagram shows some info about fuel cut off and that P0139).
For a Cat fault, you may notice the rear 02 fluctuating at a higher rate, (0.2v-0.8v-0.2v etc) when it should be at a more steady 0.6v-0.7v range. Of course noticing some of this will not be conventional with a hybrid, since youre not always going to be running on the gas engine. But under heavy throttle it should switch over to the gas engine for more power.

The hoses you mentioned may be part of the issue here though as well, without the vacuum hoses all sealed up and routed properly, that will throw off the ECM's self test results. It may cause the rear 02 to be below the 0.6v threshold that is called for in the code criteria. Looks like another thing they may need to correct, if it wasnt like before going in to them,

The p0401 looks like the ECM measuring a reference pressure(so some stored predetermined amount) to intake manifold pressure during EGR valve closed and the manifold pressure. This test is run during Fuel Cut off of more than 10sec as well, so thats interesting, these codes are being tested for and setting for this same period of fuel cut off.

But another issue is that the wiring harness on this vehicle has been connected and disconnected how many times now. Thats not at all a helpful thing. Who knows how much the harness connectors have been rough handled, pulled and pushed on. Theres so many factors here.

Another thing about monitoring the rear 02, is if voltage is fluctuating 0v to 1v roughly, when the frequency increases the ECM will eventually set a P0420, Catalyst low efficiency Bank 1. Which may need a certain number of drive cycles to set, thats when you'll know the cat is failing. Looking a little closer at the P0139 criteria, states the code is set "when the switching time is bigger than the limit value". Thats an increase in frequency of the rear 02 voltage fluctuating, so the P0139 might set before the P0420 Cat Efficiency code is set.
Thats something we would be looking for when diagnosing a cat failure, voltage swings such as the waveform below, or if its bad enough and causing drivability issues, a back pressure test to see if the cat has melted down and is restricting the exhaust.

I don't see any pressure differential sensor for the EGR valve in the wiring diagrams, but I do see a 4 wire MAP sensor which is what they are using for intake manifold pressure measurements, what are the wire colors on that sensor?
The EGR valve has a position sensor built into it as well, so it will have 5 wires.