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 it permissible for a Muslim woman to leave her house adorned and without wearing her proper Islamic attire?
A Muslim woman isn't permitted to leave her house adorned and without wearing proper Islamic attire. And Allah Knows Best.
What should a sick person, who is incapable of fasting, and who didn`t make up for missed fasting days, do?
One who has missed fasting days is obliged to make up for them, but if he/she wasn`t able to because of an incurable disease , or old age, then he/she has to pay a ransom which is feeding a needy person for each of the missed fasting days. And Allah Knows Best.
What is the ruling on giving my mother a gift at "Mother's day", is it Halal or Haram?
Praise be to Allah the Lord of the Worlds.
There is nothing wrong with doing that as long as it isn`t done in imitation of the disbelievers. And Allah The Almighty Knows Best.