To answer the original question:

It depends how the Code memory is stored, and what the processor is.

FLASH devices cannot be programmed and read at the same time, so this is a problem if your code is being exectuted from the same device. If you have 2 separate FLASH devices for code, you could run from one and program the other.

It's also possible to run code from the controller's eprom while programming extrnal flash.

What to send over the serial line:
Binary is the simplest, because it does not have to be decoded! It theoretically could be transferred on one large block, but some synchronizing would help a lot.