Is my wife cheating or just a flirt? Should flirting with others be allowed when married?

Flirting should be a serious red flag. If you allow flirting in your mind (meaning if your wife allows flirting into her mind) then she is allowing temptation. That temptation will fester and become actions. It is one thing to give a compliment or receive them but when you leaving those doors open and conversing with them through email for no reason, then that is a much bigger problem. Honestly, she is probably cheating on you or about to at least.