I look forward to learning about the solution to your shifting problem. I have a similar problem with my 1010, except that it shifts fine when the engine is cold, but then is very hard to shift or won't shift when the engine gets hot.
I suggest that you get a John Deere 1010 Repair Manual and a Parts Manual. You can either get them off ebay or directly from a John Deere dealer. They are indispensible.
As for the radiator question, I just took the radiator out of my 1964 1010 gasoline RUS, which may be completely different from your U diesel. I also had to replace the seals on my steering gear when I was in there. I ended up having to remove the hood, steering column, front axle, and steering gear to get to the radiator. Hopefully yours will be easier.
Sounds like you got a great deal, by the way, even if you have to put a bunch of money into it. My RUS has no loader, but I wish it did.