Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
What is the ruling if hemorrhoid blood exits after completing ablution?
If this blood is exiting from outside the anus (due to the hemorrhoid protruding), it does not invalidate ablution, because blood exiting from the body from other than the two orifices does not invalidate ablution. If it exits from the anus (meaning from inside it), it invalidates ablution, and one must perform istinja' from it, wash the area of impurity, and repeat the ablution.
However, if this blood exits continuously such that no time remains sufficient for purification and prayer without it flowing, then it takes the ruling of urinary incontinence (sals al-bawl). One then cleanses from it after the time for each prayer enters, performs ablution immediately thereafter, and performs the obligatory prayer immediately. There is no liability upon him after that if something of it flows, and he may pray as many voluntary prayers as he wishes. If he wants to pray another obligatory prayer, he must cleanse himself and perform ablution. And Allah the Almighty knows best.
Is it valid to share in the 'aqīqah by contributing a seventh share of a camel or cow?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Yes, it is permissible. If a group of people jointly share in the slaughter of a camel or a cow on behalf of seven individuals, this is valid — regardless of whether all of them intend the 'aqīqah, or some intend the 'aqīqah, others the uḍḥiyyah, and others simply the purchase of meat. And Allah Almighty knows best.
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.