Is it possible to change status from a J-1 trainee to H-3 before the 18 months training under J-1 expires? Is it possible to do it with the same employer? Please note that the J-1 in this case is not subject to the 2 years foreign residence requirement.

You can not change status if you are subject to 212(e) the 2 year home residency requirement.

In this case, the person holding the J-1 trainee status is not subject to 212(e) or 2 year home residence requirement. My question is whether he can change to H-3 which is another training visa (after having done a training under J-1 with the same employer that will sponsor him for the H-3).

why aren't you going back to your own country, as you promised, instead of attempting to change to some other phony status to continue working in my country? Go home.