Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?
The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.
What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
What is the amount of food to be given to a needy person for fidyah and kaffarah?
The amount of food to be given to a needy person is 600 grams of wheat or rice.
According to the Hanafi school, it is permissible to give the monetary equivalent instead, and this is the ruling issued by the General Iftaa` Department.
When is the prayer considered *adāʾ* (performed within its time): upon rising from the bowing (*rukūʿ*), or after rising from the prostration (*sujūd*)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Whoever catches one complete rakʿah within the prayer's time has caught the prayer as an on-time (adāʾ) prayer. The rakʿah is complete when the worshipper raises his head from the second prostration (sajdah), even if he has not yet reached the degree of standing in which the recitation is valid.
It is stated in Nihāyat al-Muḥtāj (Vol.1/p.378): "What is meant by the rakʿah is the completion of all of it, with its two prostrations." End quote.
And in Ḥāshiyat al-Shabrāmallisī: "(His saying: 'a rakʿah') means that he raises his head from the second prostration, even if he has not reached the degree in which the recitation is valid." And Allah Almighty knows best.
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Note on terminology: Adāʾ refers to performing the prayer within its prescribed time, as opposed to qaḍāʾ (making it up after its time has ended).