What is the ruling on washing najis clothes with soap and rinsing them with water before washing them with other clothes, and is it necessary to inform one's father about this if there is embarrassment?

You are not obligated to inform your father about the impurity of his clothes. The default state of clothes is purity. The impure part of the clothes should be washed before placing them with pure clothes in the washing machine, because placing impure clothes in a small amount of water renders it impure. Be cautious of falling into obsessive doubts (waswasah).