I suffer from chronic asthma which requires ventoline inhale; which is an air causes the risperatory track/airways to expand. Am I permited to use it during the daylight of Ramadan? And does break one's fast? Noting that I can't go an entire day without using it?
Taking inhaler through the mouth or the nose breaks one's fast, because it is meant to reach the lungs which are part of the coelom. Therefore, whosoever use inhalers for couple of days is obliged to preserve his fasting and make up that day later. As for the ones the ones who need to use it every days are obliged to preserve his fast and feed a poor per day and the cost of feeding one poor is estimated (60) piasters for the year (1421AH). And Allah Knows Best.
What is the ruling on someone who perform their prayer after finishing it because they believe they missed a Rak'a, a prostration, or that they did not perform it correctly (thinking their prayer was invalid)?
If he was sure that his prayer is void then reperforming it is an obligation along with figuring out the reason of invalidity so long as this wasn't out of uncertainty. And Allah Knows Best.
Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams (unmarriageable kin).
What is the ruling on a Muslim who slaughters an animal while being in a state of Janabah (ritual impurity)?
All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upo all of his family and companions.
Slaughtering is permissible for the ritually impure man and the menstruating woman because ritual purity isn`t a condition for the validity of the slaughtering. Moreover, it is permissible for the uncircumcised person to do the slaughtering as well. And Allah The Almighy Knows Best.