1. I am worried about auto inoculation specifically when it comes to shaving. Is it safe to shave the genital area when I have a cold sore on my lip? For clarification, I am worried about micro cuts in the genital region and accidentally touching my cold sore and then spreading it to the genitals.
This is not a concern. Once you have a well established infection in one location (which you do), it is highly unlikely you would get it in a new location on your body
2. I am interested in a profession where shaving is required frequently. Is it safe to shave my face when I have a cold sore? Concerns are for the same reason in my first question. I am worried about nicking the cold sore and then spreading it to other parts of my face?
It is safe to shave, yes, most definitely.
3. How likely is it that this could spread to the eyes?
I don’t have concerns about you spreading this to your eye but sometimes the virus travels on the top branch of the trigeminal nerve, where the cold sore virus lives, and infects the eye. But without any ocular symptoms, I wouldn’t worry about this at all. It is not common.
The main take away here is that once you have this virus well established (and you do), it is extremely unlikely that you would spread it to a new location on your body