Yes, but takes longer for conclusive results than the blood antibody tests. It's the one standard test that really does require 3 months (or close to it) for definitive results.
A oral fluid test is an antibody test
Yes -- for sure if it was a "duo" or "4th generation" blood test, and virtually conclusive for a plain antibody test. If it was an oral fluids home test, another test at 3 months is needed to be conclusive.