By how many degrees does a broken bone have to be offset to have to get re-broken?

Not always. Based on degrees of offset. Fractures need to be "set" if there is more than a 2 mm. Gap or obviously malaligned. The idea is to get the fragments as close together as possible, along with as normal alignment as can be obtained. If healed already, then revisiting the fracture is done only if it is causing some other problem.
Depends. The acceptable alignment of a fracture depends on the particular bone, location of the fracture, amount of msalignment and the age of the patient which affects the amount of remodelling which will occur. So without more specific info it would be difficult to answer your question.