Vasomotor rhinitis. This form of rhinitis, runny nose, is non-allergic and not due to inflammation either, and yes, sexual arousal can be a cause. So if you were referring to feeling warm, flushed 9some other common bodily sensations post coitus) and stuffy during and after having sex, then this is most likely the cause.
Sexually transmitted. This is possible, particularly if you have contacted a sexually transmitted disease. This issue should be addressed by an internist or if you are a female by a gynecologist.
No. Regular sex (penis into vagina) should not cause flu-like symptoms. Flu-like symptoms can occur when some inflammatory substance, virus, or bacteria spreads through the body, causing fever, aches, fatigue, nausea, and sometimes vomiting. It would be very unusual for such symptoms to be from sexual activity, but if such symptoms repeatedly occurred with sex, a doctor's evaluation is needed.