What exactly should I ask the father of girl I want to do mutaa with? The father is American and a christian, should I ask to simply date her? Or explain mutaa to him? What is it that i should ask permission for?

You can tell him that in my religion I cannot have a relationship with or befriend a girl unless I do this verbal contract, which is a temporary marriage contract. It means I respect her and recognize that she has rights and I have rights, and that I would not be violating the law of God. You don’t have to go into specifics of what you want to do, but just explain this to him and if he approves you can do the Mut’a contract with her.