What is the ruling of Islamic Law if the follower (maʾmūm) stands up for the third rakʿah out of forgetfulness while the imam has sat for the middle tashahhud?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If the imam sits for the tashahhud while the follower (maʾmūm), out of forgetfulness, stands up for the third rakʿah, he is obliged to return to his sitting position, since following the imam is obligatory. In this case, no prostration of forgetfulness (sujūd al-sahw) is required of him. If, however, he fails to return, his prayer is rendered invalid.
Shaykh al-Islām Imām Zakariyyā al-Anṣārī, may Allah have mercy upon him, states: "If the follower stands up alone out of forgetfulness, he is obliged to return, since following the imam is obligatory... If he does not return, his prayer is rendered invalid, due to his departure from what is obligatory." [Asnā al-Maṭālib,Vol.1/P.190] And Allah the Almighty knows best.
What is the ruling on swimming while fasting?
Swimming while fasting is disliked (makruh) due to the risk of water entering the body cavity (jauf) through the nose, ears, or mouth, which would break the fast.
Ramadan is a month of tasbeeh (glorifying Allah), not Sibaha (swimming).
What is the ruling on someone who is uncertain whether he omitted one of the ab‘āḍ of the prayer, such as being uncertain whether he omitted the first tashahhud?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The ab‘āḍ include the middle tashahhud and the qunūt in Fajr prayer — for which sujūd as-sahw (the prostration of forgetfulness) is prescribed if left out. Whoever is uncertain, during the prayer, whether he omitted one of these elements, should proceed on the assumption that he did not perform it, and should perform sujūd as-sahw accordingly.
It is stated in ‘Umdat as-Sālik (p. 63): "If he is uncertain... whether he omitted a specific ab‘āḍ element, or whether he performed sujūd as-sahw, or whether he prayed three or four rak‘ahs — he proceeds on the assumption that he did not do it, and performs sujūd as-sahw."
The original principle (al-aṣl) for someone uncertain about [having performed] the first tashahhud is that he assumes he did not perform it. And Allah, the Exalted, knows best.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.