Of course, in order to copy the bank switching code to the other banks you need to switch to them first. Which means that this copying can only be done by the ROM.
Only one bank can be active at a time, so the only RAM you can run the code from is the same bank that you're trying to load code into. Slight chicken-and-egg problem.
No, that's not how these systems work. The bank switched RAM is switched in banks of 16K, so you can keep 48K 'constant' while the other 16K is switched as a window in the 256K.