Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
The imam recited a verse containing a prostration of recitation (*sajdat al-tilāwah*) but did not prostrate. After the imam's *salām*, a follower (*maʾmūm*) performed a prostration of forgetfulness (*sujūd al-sahw*) because the imam had not performed the prostration of recitation. Is it permissible to go against the imam in this way?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is no sin on the imam if he omits the prostration of recitation (sujūd al-tilāwah), even deliberately. The maʾmūm (the one following the imam) is obliged to follow the imam in the prayer, so if the imam does not perform the prostration of recitation, it is not permissible for the follower to prostrate, because the imam is appointed only to be followed.
If the follower introduces a prostration after the imam’s salām but before his own salām, in a case where there is no prostration of forgetfulness (sujūd al-sahw), his prayer is invalidated if he knew it to be prohibited. Otherwise, he is excused on account of his ignorance and his prayer is not invalidated.
It is stated in Ḥāshiyat al-Imām al-Jamal ʿalā Fatḥ al-Wahhāb: “His statement: ‘So the omission of the hayʾāt (the non-essential sunnahs of the prayer) is not compensated for by prostration; if one prostrates for any of them deliberately, his prayer is invalidated, unless he is excused due to his ignorance.’”
Since this issue is one that is hidden from the laymen, as it is among the subtleties of knowledge, and the laymen are excused for ignorance of whatever is of such a nature, and their act of obedience is not invalidated because of it, it may be said that the prayer of one who has not acquired knowledge of this issue is not invalidated, with the necessity of learning the rulings remaining. As for after knowing it, the prayer is invalidated. And Allah Almighty knows best.
Someone wiped over his shoes after wearing them in a state of purity, then took them off and prayed without them?
Ablution is not invalidated by taking off leather socks or shoes after wiping over them. However, whoever takes them off after wiping must wash his feet only. If he prayed without washing his feet, he must wash his feet and repeat the prayer. And Allah the Almighty knows best.
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.