If this person was on daily valtrex for suppression, transmission with four intercourses is unlikely. Semen is not a great carrier of virus, no. The coughing means nothing in terms of herpes. Your symptoms are not concerning to me.
There are some false positives on the herpes IgG tests, your doctor is correct, but not for the reason he states. By the age of 45, 95% of the population in the US tests positive for the virus that causes chicken pox and no where near 95% of the patients I see test positive on the IgG test – we don’t know why people test falsely positive.
There is a great test called the herpes western blot that never yields false positive results and if you decide to test, that’s the test you need. I could help you with that.