It does not test for antibodies at all. That is a different test and not being given in these numbers. These tests are looking for active infection with the virus itself. The people testing positive are infected, whether they are symptomatic or not.
You are incorrect. There can be late false positives not indicative of infection at all. Please learn about PCR
You are incorrect. There can be late false positives not indicative of infection at all. Please learn about PCR