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.
Does the intention (niyyah) of the principal (muwakkil) suffice for the sacrifice (dhabḥ) of the udhiyyah?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
If a person appoints an agent (wakīl) to perform the slaughter (dhabḥ) on their behalf, the intention (niyyah) of the principal (the one appointing the agent) suffices, and there is no need for the agent to have this intention himself. In fact, even if the agent is unaware that he is performing the sacrifice on someone's behalf as an udhiyyah (sacrificial offering), this does not affect its validity. And Allah, the Most High, knows best.
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.
What is the ruling on someone who intends to travel to Jeddah for two days — must he enter iḥrām from his place of residence [i.e., before passing the mīqāt]?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Whoever is firmly resolved in his intention to perform ‘Umrah before setting out on his journey, it is unlawful for him to pass beyond his country's mīqāt (the designated boundary point for entering the state of iḥrām) without entering iḥrām. If he does so, he must return to the mīqāt to enter iḥrām from there, provided he has not yet begun performing the rites of the pilgrimage — in which case nothing further is required of him. However, if he does not return, he must offer a sacrifice (sheep/goat), to be distributed among the poor of the Ḥaram.
It is stated in at-Tanbīh fī al-Fiqh ash-Shāfi‘ī (p. 71): "Whoever, intending to perform the rites [of ‘Umrah or Ḥajj], passes beyond the mīqāt and enters iḥrām only after it, must offer a sacrifice. However, if he returns to the mīqāt before beginning any of the rites, the sacrifice is waived from him."
Accordingly: whoever is firmly resolved in his intention to perform ‘Umrah before setting out on his journey must enter iḥrām at the mīqāt of his country through which he passes. Whoever fails to do so, passing beyond the mīqāt without entering iḥrām, and does not return, has sinned, and must repent to Allah, the Exalted; and he must offer a sacrifice, to be distributed among the poor of the Ḥaram. And Allah, the Exalted, knows best.