Is it required for a discerning child to make the intention at night if they wish to fast?
If a discerning child wishes to fast during Ramadan, they must make the intention at night because intention is one of the pillars of fasting, even though fasting is not obligatory for them.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible for one who sacrifices on behalf of another with his permission to eat from the sacrifice with his permission, and he takes his place in distributing its meat.
It is stated in Hashiyat al-Jamal 'ala Sharh al-Minhaj (Vol.5/P.262): 'If a person sacrifices on behalf of a living individual with his permission, does he take the place of that individual in distributing the meat, because permission for the sacrifice implies permission for distribution, or does it depend on explicit permission? There is room for consideration, but the former view is not far-fetched.' And Allah Almighty knows best."
How to perform the witr prayer in terms of connection (wasl) and separation (fasl)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Witr prayer has several forms that vary in terms of virtue:
The First Form: Separating every two units (rak‘ah) with a Tashahhud and a Taslim (salutation). This is superior to connecting the units, even if it is only a single rak‘ah. This is based on the Hadith of ‘Aishah (may Allah be pleased with her): 'The Messenger of Allah ﷺ used to pray eleven units between the end of the ‘Isha prayer and dawn, performing the Taslim after every two units and performing Witr with a single unit.' (Related by al-Bukhari & Muslim).
The Second Form: Connecting the units with only one final Tashahhud at the very end.
The Third Form: Connecting with two Tashahhuds—meaning reciting the Tashahhud before the final unit without performing the Taslim, then standing to complete the final unit. This form is considered the lowest in rank so that the Witr prayer remains distinct from the obligatory Maghrib prayer, as stated in the Hadith: 'Do not make the Witr resemble the Maghrib prayer.' (Narrated by Al-Daraqutni, who stated its narrators are trustworthy).
It is stated in Bushra al-Karim Sharh al-Muqaddimah al-Hadramiyyah: 'It is permissible to connect [the Witr] with one Tashahhud in the final unit—which is better—or with two Tashahhuds in the last two units, as both methods are established in Sahih Muslim from the actions of the Prophet ﷺ. In the connected method, more than two Tashahhuds are prohibited. Furthermore, separating (al-Fasl) is better than connecting (al-Wasl) if the number of units is the same, because the Hadiths supporting it are more numerous and it involves more devotional actions.' And Allah the Exalted knows best."
Do ear and nose drops break the fast?
Yes, ear and nose drops break the fast because both the nose and ears are open passages to the body cavity.