To answer your first question, Dayquil (main active ingredients: acetaminophen (tylenol), dextromethorphan (for cough suppression) and phenylephrine (helps vasoconstrict nasal passageways and 'dries out' a runny nose) will not adversely react with or really affect in any way your Allegra (antihistamine) nor your Advair (inhaled fluticasone (steroid) and salmeterol (bronchodilator, long acting). Feel free to take some Dayquil to see if it helps improve your symptoms, but do not take a dose that would deliver more than 4g acetaminophen in one day otherwise you may damage your liver. It is difficult to say what you are ill with, but most likely it sounds like a viral upper respiratory tract infection and thus would not be treated with antibiotics and the best treatment is to rest and take in ample amounts of fluid. That being said, if your symptoms worsen or if fevers develop/persist it could be a sign that your are developing a bacterial infection and you should go in to see your doctor
at that point. Also, if your congestive symptoms last longer than a week or so it would be reasonable given your allergy history to go in to see your doctor and inquire about a nasal steroid such as flonase which can reduce inflammation in the nasal passages and improve congestion and cough.