Is the repentance of one who masturbated with another man accepted, and what is the method of acceptable repentance, and how does one know that his repentance has been accepted, and is what he did considered sodomy if the other person only touched his penis with his hand?

What is mentioned in the question is more heinous than mere masturbation and could lead to sodomy. It is obligatory upon the perpetrator to repent, regret, and seek forgiveness. Sincere repentance is that which fulfills the following conditions: sincerity, abstaining from the sin, regretting it, resolving not to return to it, and that it occurs before the death rattle and before the sun rises from the west. This act is not sodomy, but it is heinous and a gateway to it.