HSV is not spread through laundry, generally speaking. If you have genital HSV 2 and you use a towel or washcloth and you are shedding virus and someone immediately after you uses the exact same towel or washcloth on their genitals, no soap involved, transmission is theoretically possible but so unlikely
What were the values of your IgG test results?
We know that 95% of the time, HSV 2 is genital
YOu would not transmit HSV 2 by rubbing your eyes and touching someone else, no.
HSV 2 is very rarely oral, but if you get the white spots again, you should have them swab tested to see if they are herpes. I would be surprised if they were.