Is housemaid right hand possession?
All perfect praise be to Allah, The Lord of The Worlds. May His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Housemaid isn`t right hand possession. Rather, she is a free woman because she is just hired to deliver a service. And Allah The Almighty Knows Best.
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
Is touching a non-Mahram woman invalidates person's ablution?
Touching non-Mahram woman is prohibited, and according to the Shafiee's, it invalidates the ablution. And Allah Knows Best.
Are all the children of a suckling woman considered brothers to the strange child whom their mother had breastfed, or just to the one who had suckled with him?
All the children of the woman who had suckled a strange child are considered his/her brothers and sisters.