What is the ruling on a Friday sermon in which the khaṭīb did not explicitly exhort the congregation to be conscious of Allah (taqwā) in both sermons, but sufficed with commanding them to obey Allah and refrain from disobeying Him?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
For the Friday sermon (khuṭbat al-Jumʿah) to be valid, certain essential pillars (arkān) must be fulfilled. Among these is the exhortation to be conscious of Allah (waṣiyyah bi-taqwā Allāh), which must be present in both sermons. Alongside this pillar, the praising of Allah (ḥamdallah) and the sending of blessings upon the Messenger of Allah ﷺ are equally required.
Shaykh al-Islām Imām Ibn Ḥajar al-Haytamī, may Allah have mercy upon him, states: "These three are pillars in each of the two sermons, because each sermon is independent and separate from the other." [Tuḥfat al-Muḥtāj,Vol.4/P.447]
It is not a condition that the exhortation be expressed in any specific wording, nor is it required that the word "taqwā" itself be used — such as saying "I exhort you to be conscious of Allah." Rather, this pillar is fulfilled by any expression that contains a command to obey Allah the Almighty and to abstain from what He has prohibited.
Imām al-Khaṭīb al-Sharbīnī, may Allah have mercy upon him, states: "The third pillar is the exhortation to taqwā... The specific wording of this exhortation is not required, according to the most correct view, because the purpose is admonition and the urging of obedience to Allah the Almighty. Therefore, any expression that conveys admonition suffices — whether long or short — such as: 'Obey Allah and be ever mindful of Him.'" [Mughnī al-Muḥtāj,Vol.1/P.550]
Accordingly, what the khaṭīb has done — by commanding obedience to Allah and forbidding disobedience to Him in both sermons — is valid and sufficient. And Allah the Almighty knows best.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
It is permissible for someone who sacrificed on behalf of another with their permission to eat from it with their permission, and they stand in their place (act as their representative) in distributing it.
It is stated in Nihayat al-Muhtaj ila Sharh al-Minhaj (Vol.8/P.141): "And he—meaning the one sacrificing on his own behalf, provided he has not apostatized—has the right to eat from a voluntary sacrifice and his sacrificial gift; rather, it is recommended. As for an obligatory sacrifice, it is forbidden for him to eat from it, whether it was designated as such initially or as a liability in his dhimmah. And excluded by what has preceded is if he sacrifices on behalf of another, or if he apostatized, then it is not permissible for him to eat from it, just as it is absolutely impermissible to feed a disbeliever from it."(The text I found indicates the impermissibility of eating) ???
And it is stated in Hashiyat al-Jamal ‘ala Sharh al-Manhaj (Vol.5/P.262): "If a person offers a sacrifice on behalf of a living person with their permission, does he act as their representative in distributing it—on the grounds that permission to sacrifice is permission to distribute it—or does it depend on their explicit permission? This requires consideration, but the first view is not far-fetched." And Allah the Almighty 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.
Is Fajr (dawn) Sunnah a confirmed one?
All praise be to Allah The Lord of The Worlds Of course, it is a confirmed Sunna which the Prophet (PBUH) used to offer in a regular basis. And Allah Knows Best.