Radiator questions

Can you solder or braze it even if it's leaking from the core?
You can cut and close off the offending tube(s) while assembled IF it is leaking well up into the rad core, and not at the point where the tube connects to the top or bottom plates of the core. You have to cut some of the surrounding fins to get a clear area to solder. Silver solder is used so higher than normal solder iron temps are needed.

If it leaks right where the tube connects to the top or bottom plates, then it usually has to come apart.

If you see a lot of signs of weeping (like a number of greenish spots all over), then the tubes are just getting corroded and thin and a new core will be needed eventually, and tube repairs become iffy.