What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.
What is the ruling on brown discharge before the menstrual period? Is it considered part of menstruation (Hayd)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Dusky discharge (Kudrah), reddish discharge (Humrah), and yellowish discharge (Sufrah) are all considered menstruation (Hayd) if they occur during the time of the menstrual cycle. If their duration exceeds a day and a night, and the period persists from the first sight of the discharge until the cessation of the menstrual blood—provided the total duration does not exceed fifteen days—then all of it is menstruation. However, if the duration exceeds fifteen days, then the discharge is not considered menstruation, but rather chronic irregular bleeding (Istihadah). And Allah the Exalted knows best.
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.
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.