Can you give yourself genital herpes from your own cold sore?

Technically..: Yes, you could if the sore was open and shedding the virus, and you touched the sore on your mouth and then right after touched your genitals.Although this is more of a technicality,studies show that it is becoming more common than it used to be in the last decade.I would generally avoid immediate genital contact after touching an open cold sore as a rule to be safest.

No: This is not possible. Cold sores and genital herpes are different entities

NO: Your immunity confines it to 1 place. This does not happen. Notice that you are shedding herpes virus BETWEEN attacks when you have no symptoms. In case of severely impaired immunity herpes virus may spread widely, not just to genital skin.