So yes, syphilis, HPV and herpes can be transmitted by rubbing unclothed genitals together. They do take friction, usually the kind associated with intercourse.
Have you been vaccinated for HPV? It won't treat or prevent an infection you already have, but if you have, it will significantly reduce your chances of getting an infection from this encounter.
There is no test for men for HPV, unfortunately. The only test for women is done in conjunction with the pap smear.
Syphilis is uncommon in most developed countries - the US, UK, Canada, countries in Europe, etc. I wouldn't worry too much about this.
You can test for syphilis at 6 weeks if you feel a need to.
Herpes is trickier, since many countries don't like to do the blood testing, and to be fair, they aren't all that accurate. The hsv1 IgG blood test misses 30% of infections, the hsv2 IgG blood test misses 8%, and has a fair number of false positives.
It's easy to get a blood test in the US, but not easy in Canada, the UK, Europe, etc.
Since this is a new relationship, it's reasonable for both of you to test before any other sexual activity takes place. You can ask for hsv testing, but don't be surprised if it isn't offered if you aren't in the US - and even some doctors/clinics in the US don't like doing them.
He says he tested in Feb. He could have had partners since then, or had partners very close to that testing that wouldn't appear on those test. He can test again if he wants to be intimate with you.
Take a breath. Your chances are probably low. Since you mentioned health anxiety, have you ever talked to anyone about that? Maybe now is a good time. There are great treatments for it.