We don’t have any statistics on transmission of HSV 1 to someone else, either through kissing or through giving or receiving oral sex. Sorry. People who have symptoms with an HSV 1 infection likely shed more than those who don’t have symptoms. People with genital HSV 1 shed virus far less often than people with oral HSV 1.
The situation with your mom presents no risk of transmission. The skin of the arm is too thick for virus to penetrate and infect someone who kisses it with a cold sore.