Not sure what's going on here. Car isn't running.
Service councilor 26 is listing for '52, not '50.
There's a complete Ultramatic (700pps) service bulletin covering all years in the literature archive.
Best to start there.

I suspect that running them together is suggested because the internal rad mounted also warms the fluid at cold start-up.
With a front exchanger only, may not get to operating temperature and cause sluggish shifting for a while or for the duration. No direct experience, just engr'g practice.

Personally, if your trans is 'overheating' or requiring additional cooling beyond factory, it's due to service/load changes, towing, hilly, etc. or fault in the trans. If that's the case and not just exploring available period options, I'd first opt for a thorough machine cleaning and fluid change depending on the maintenance history. Particulate settles out in the radiator trans coolers, too. Just servicing the trans won't get that and it just recirculates leading to excessive wear.

They did a lot of things in those days that today don't make much sense on inspection. You'll see a lot of new items that come and go each year as the technology was being perfected. Automatics are in their infancy in this period. P was a little more sluggish in that regard with sales lagging expectations while sticking to outmoded, but proven, designs compared to competition. They were more conservative and high end historically.

Be advised that combo weighs in around 1000 lbs for 135 HP and will be quite front heavy in an "A" and be longer than a flat 8.
May require frame mods, sectioning, etc.
There are examples on the internet, some turbo charged.

Not sure exactly what the plastic is, mostly tenite and PVC in that era.
Tenite is known for the white film degradation. See link below.
I was surprised the surface wasn't more disturbed requiring body work, if that occurred, so maybe something just contaminating. Must have been living a sedate garaged life. When they sit out in the sun with the windows up and the internal temp can reach the Tg of some of the plastics, they'll have no choice but to move/crack, etc..
I don't know what the mechanism of bending/distortion was but it's either from plastic shrinkage along residual molding stress lines or swelling/deformation of the door card and pushing/tensioning on the arm rest(creep). I doubt 'over' tightening would do much except crack the parts. Dimensional stability of early plastics was not that good/not scientific. '50s is the infancy of 'modern' plastics. Today, they model out the injection molding, custom formulate the material, etc. in the design of the parts where in those edisonian daze it was just make parts best guess and throw them out if they weren't right, aka trial by error.

Anyway - looks nice and good work all around.

https://en.wikipedia.org/wiki/Tenite

go to the "main menu" on the left hand side and click 'model information'. Select your vehicle, click on the model link and down the bottom should be a paint chip link.

Plastics often have mineral fillers, like calcium carbonate, etc. and that might be what the white stuff is if the surface was oxidized, but can't tell from a pic. ie, sign of degradation likely. It might be advised to paint the plastic bases if they're not cosmetically good/preserve. There's special spray paints for plastics, may need a surface prep/adhesion coat.

I'd check the filler foam you select on a small section first so it doesn't soften or otherwise destroy the plastic. Could end up with goo. Shouldn't need filling, if they're sound.

I'm not sure where the deformation is from the pics and heating may not be advised as they were made in a mold and to heat them without could cause more distortion. It's also 70 years old so the plasticizers etc, may not be good and may crack, burn, etc. instead. You'd need a temperature controlled oven set to anneal the specific plastic and not just a heat gun which will ensure uneven heating.

Fabric backed vinyls come in different thicknesses, so need to make sure it's the right kind so it can be formed properly.

Goes to prove people will buy anything if the price is right.
In seven decades, I don't recall ever buying something I didn't know what it was, but I'm still hopeful! Someday... "Never know when I might need one, whatever it is."

Well. someone confirmed what I warned of. Parts in the pans is never good news. No telling where they been. It gets expensive going into surgery for a minor noise. If they were solids, they probably would adjust out. Hydraulics on a flat head is kinda weird. Overheads you can adjust/service them easier. Then some of these cars don't have any oil filter or partials, excess wear and gunk accumulates. That stuff in the pan will get "lifted" by modern detergent oils, and away she goes making noises. I made sure I cleaned the pan (soaked in kerosene) before I ever started the 288. There was something that turned akin to goo, not oil. Imagine if I had tried to start with that going on? Took lots of grief, but it still had a slight lifter noise, but it's got solids. Lazy to adjust and sitting again anyway.
Hope others that see this and going to buy one, notice the tell tales. Welds on the radiator support, homemade parts, noises, etc. And those that let them sit, take notice and regularly exercise and change fluids.

That's of the founding. We've passed the double point, they've been de'funk't more years than they were in a business! I don't think it matters now?
Just reminds me how old I'm getting, ie, the relic I'm becoming...Oh no, not another one without my project finished. Like the recent lost moon shot, after 50+ years they couldn't do it again? Someone lose the notes or what? No app for that? It's still on TV so someone messed up big time. But some OH(the homeland) guy is going to try to make Ps? Maybe he'll have better luck, longer success.

I think that's a valve seal where you see the extraneous 'shim' Retainer probably broke at some point and rather than replace/change, they attempted a bandaid. Of course, begs; what happened to the pieces? Could be a clue. Without history, at a loss.
without disassembly, no history, etc. you might try a dose of "valve ease" or similar and let it run some miles and see if it quiets any or not. It'll go out with the next oil change - no risk. If still there after add, then it probably need a bit of surgery to cure what ails it.
Depending on the mileage and condition, it can be anything. You're only seeing the obvious, odd man out, out of normal thing - related or not. You might be able to better isolate which one is tapping, with a stethoscope. Can't tell from the videos. Sound seems to come from some other spot. But can't rely on a sound recording like that. There are sonic probes for that but not common DIY.
Might be one of those don't worry about it till it gets worse items. Got a 1:3 chance, it'll go away.
It could be a costly chore to cure a small noise on something one may not drive much. Plus no guarantee something else won't crop up during surgery. The "oh, while I'm in here I might as well..." cost accelerator. There comes a point where it's a rebuild bite the bullet. You may not like the results in the pocket book.

This has probably been covered before, so search the site for more info.