Can the bytes that make a word be handled individually?
Yes you can. For example, the Texas Instruments MSP430 line are 16 bit processors, and have 16 bit words. A variety of registers are 16 bits long but can be addressed as 8 bit registers as well. It's memory bus is 16 bits wide though. You could add 8 bit bytes to memory without issue. And use both bytes of a memory location independently or together if you want.