If I understand your question, if you wish to do something to one IO pin do you have to manipulate it through a full byte, the answer is yes and no.
That is each IO pin is associated with an IO port. Each IO port has up to 8 pins associated with it (some may have less). So in your example the pin PC0 is saying pin 0 or port C. And looking at the docs Port C only has actual IO pins on it.
The IO Port C has 3 IO registers associated with it.
PORTC - You can output the value per pin here.
DDRC - Pin directions - Are the pins input or output
PINC - Read in the value of each of the pins.
Now you can set the value of all 8 logical pins at once by doing something like:
PORTC = 0
That will set all 8 pins to value 0 (off).
But the processor also has bit manipulation instructions that allow you to set/Clear... one specific bit.
Got to run
Kurt