Typically a throat culture
only looks for strep throat
caused by one kind of bacteria. However it is still possible to have a sore throat
and have a negative culture.
With his symptoms and history of kissing, I would also be concerned about Mononucleosis
or in short Mono which is viral infection that often presents with sore throat, fever, headache, fatigue, etc. There is a rapid test called the Mono spot that could be performed.
For sinus infections, typically he is going to have a purulent (yellow-greenish) nasal discharge and pain over his sinuses (located on cheeks and forehead). If this is the cause, he may have a sinus infection however it is unlikely that she gave him the sinus infection but more likely gave him a viral upper respiratory tract
infection and his sinuses were not able to drain which can lead to an infection. Many sinus infections will resolve on their own in 7 days however if they persist or continue to worsen, then antibiotics are needed at that point.
Good luck and take care :)