You could get them on your face, and possibly in your mouth, though that's uncommon.
"Primarily a skin infection, molluscum contagiosum lesions can also occur on mucous membranes, although involvement of the oral soft tissues is uncommon. Oral lesions occur most frequently on the lips, tongue, and buccal mucosa." (www.cdho.org/Advisories/CDHO_Factsheet_Molluscum_Contagiosum.pdf)
You may not see symptoms for awhile. The incubation period (time from infection to time of symptoms appearing) ranges from 1 week to 6 months, but the average is 2 to 3 months. (http://www.ashasexualhealth.org/stdsstis/molluscum-contagiosum/)
Removing the lesions can cause scarring. Often, doctors will opt to leave them to resolve naturally, since they don't cause scarring, though that has pros and cons to it. Removing them can help prevent the spread, but the treatment can cause scarring.
There is an oral treatment. Facial lesions don't respond as well, if you get them on your face - https://www.cdc.gov/poxvirus/molluscum-contagiosum/treatment.html
Was he diagnosed by a visual exam? While it's likely it's molluscum, as they have certain features doctors look for, you might want to check out pics online to compare if you haven't already. If there is any question, he should get a biopsy.
I'm sorry you are going through this.