While some of those things may sometimes be true of a food allergy, they are symptoms and not what defines a food allergy. Those things can happen for a variety of reasons which are not allergy such as lactose intolerance, Celiac disease, bacterial contamination, etc. An adverse reaction to consuming something does not equal food allergy and the above mentioned causes of adverse food reactions would not respond to allergy treatments.
A food allergy is an IgE mediated immune response to a food protein that then can cause a cascade of symptoms which can be unpredictable and range from mild to life-threatening. Unlike some other adverse food reactions, the only preventative is strict avoidance.