Is prayer nullified by nose bleed?
All praise be to Allah, The Lord of The Worlds. Blood coming out of the nose, or a wound doesn`t nullify prayer regardless of its quantity. And Allah Knows Best.
Is it permissible for a woman to uncover her face while performing Umrah (minor Hajj)?
The woman while in a state of Ihram (ritual consecration) is obligated to uncover her face and hands, but at the same time permitted to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.
I prayed while I was in the hospital after a bladder surgery, and my urine was connected to me and could not be disconnected that day. I prayed after performing tayammum with my finger because I was unable to stand. What is the ruling?
After Allah grants you complete healing, make up the prayers you performed with your finger to the best of your ability, at times that suit you. And Allah 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.