I wouldn’t agree that intraoral herpes is rare. It might be rare to present with lesions in the mouth, but we know that people shed virus from inside the mouth, so whether it is a lesion or just shedding virus, the mouth is involved in the process. The virus you have lives in the trigeminal ganglia (or nerve group). This nerve group supplies activity to the entire facial area – lips, mouth, chin, nose, and even eye. How the virus decides which nerve to travel on to get to the surface of the skin is a mystery – some people always have an outbreak in the same location and some people are like you with outbreaks occurring all over the place. If you do have an outbreak in your mouth, and only swabbing would tell you whether this is a herpes outbreak or not, this is not a new infection in any way, it is a recurrence in a little different spot.
HSV 1 can be transmitted via oral sex, yes. And it does not appear to be a problem to penetrate the penis skin. Do you know if this person has ever had a cold sore? Because if they have, it is extremely unlikely that they would now get infected genitally as well.
I don’t know what the ulcers are without a swab test, preferably using PCR, to see if they are herpetic. YOu can’t go to urgent care for a swab test? If you do, try to get PCR vs. culture.