Why are bytes 8 bits? (and more)
Your question has a lot of hidden assumptions about CPU architecture.

A study of the way ancient computers were designed can reveal quite a bit about why such choices are made today.

As to why there are 8 bits in a byte comes down to technical inertia and good enough. It wasn't always like this and in the future binary as a basis for digital computing may be as out of date as vacuum tubes and electromechanical step relays are today.

Sixty years ago electromechanical was the leading edge because everything else was too unreliable. The Fujitsu FACOM 128, one is still operational, uses 5 bits as a byte to represent one decimal digit. The Digital Equipment Corp. PDP8 has a 12 bit word and does not use 8-bit bytes at all. 8-bits per byte became a de-facto standard with the IBM System 360. Because it was IBM it was easier to make yours compatible than to risk being shutout of government contracts like UNIVAC or Burroughs by not working like the IBM.

Many architectures are implemented with multiple data paths within the CPU so that the source and destination objects can be accessed at the same instant.

One of the most popular (cheapest) controller architectures is the Microchip PIC family. They have different number of bits for instruction words, ALU data, and call stacks. 

So 8-bits per byte and using the same memory for program and data storage is the least complex way to implement a CPU that yields "good enough" performance. It does this by making the fewest possible interconnects between the elements of the CPU.