Skin conditions - Someone posted the actual NFHS Rule 4 language. The bottom line is that the referee is the ultimate authority, and literally can make the decision to disqualify someone for communicable disease - even if there is a doctor's note. Now, if there is a designated, onsite, medical person - not some random nurse or doctor out of the stands, not some kid on the team whose dad or mom who is a doctor - but a person the school has officially declared is the representative of the school on medical matters for the competition, then the referee can rely on that person to make the decision.
That being said, the referee still is granted authority by NFHS Rule 3-2 to have full control of the contest and that referee's rulings are final. Why would a referee overrule the onsite medical person? It would be a very, very extraordinary circumstance and I have never personally done it, and never heard of an official doing it. But, it could be that the official judges that the expertise or judgment or partiality of the "designated medical person" is in question. For example, that person is a trainer with lots of expertise in treating injuries but with little to no expertise in diseases or skin conditions. Or, the referee finds out the "designated person" is the wrestler in question's dad or uncle, and the skin condition is oozing.
At any rate, the referee is ultimately responsible for this decision and must safeguard ALL of the competitors - and safeguard HIMSELF. You see, we can catch whatever the kid might be passing around because we have our head down on the mat looking for a pin, we are touching the wrestlers when we raise their hand, etc. Before our District Tournaments, our Head Officials reached out to their Tournament Directors and requested an onsite medical person for checking communicable diseases - precisely as an insurance policy that the referee would not make an ill-informed decision about such a condition and either letting a kid wrestle who shouldn't be on the mat, or disqualifying a kid who legitimately could have safely participated.
Another thing to keep in mind - the medical opinion that matters is the "designated onsite medical personnel". So, if the school says person A is that designated person, then it doesn't matter if you go find 5 doctors in the gym who come and render their opinion on a situation. It is not their role, and it is not their call - although, if the designated person wants to listen to their input, then that is up to the designated person. One way to think about the "designated medical person" is this - does the school formally grant this person that role, and is the school ready to go to court if someone is harmed by the actions or decisions of that person? If the answer is "No", then I don't think this is a "designated person".
Hope that was informative.