So some of the things you mention, i experience when i let it sit a lot... Not black smoke but the white smoke sometimes. I think i have a timing belt tensioner issue potentially which makes noise when cold. You could be right on the needing a rebuild, just also may be you not driving it...
It is especially annoying when the previous owners didn't even fix cheap items that are very important. However when you get it done, you should be good for several years.
Hey do you have the room to sneak a nut splitter up there on an extension and just split the nut? I guess i could look on my vehicle but i am tired already :)
Man those fins have taken a pretty serious beating! Not sure how much air can get through those. You might try to straighten them up so you get better air flow. that air flow is also keeping air from getting to the radiator.
You might lift the front wheels off, with the engine not running and turn lock to lock to see if you can get air out of it. That loud seems like air in the pump causing cavitation.
Umm your rust issue doesn't seem as bad in the pictures as you seem to indicate. But i believe you just maybe it is something you can hit with s wire wheel then some rust converter and seal those spots up.
I thought you could order from EAU there and if so can you not order from partsouq.com?