If someone had mono many years ago, can igg antibody be detected in his/her blood many years after? and can he/she infect other people by kissing?

No: Presence of antibody does not mean that you are contagious and in fact denotes protection. You cannot infect other people because you are antibody positive

YES - which: is why it is usually a fruitless test to order! and usually probably NOT unless there are acute symptoms with viral shedding from throat lymph nodes, Ebstein Barr Virus sticks around for a long time - and people can get sick for weeks or months- but it usually does not cause symptoms forever. A better test to order if you really had to - acutely is "Monospot" test or early EBV -IgA, IgM.