Which takes precedence: the 'aqīqah or the uḍḥiyyah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah is an act of worship through which a servant draws closer to Allah the Almighty in gratitude for the blessing of a newborn child. It is a confirmed Sunnah for those who are financially capable, and Allah does not burden any soul beyond what it can bear.
However, the uḍḥiyyah takes precedence, as it is a confirmed Sunnah established from the Prophet ﷺ through stronger and more numerous narrations — indeed, the Ḥanafī scholars hold it to be obligatory. Furthermore, its time is narrow and limited, expiring with the passing of the days of Eid, whereas the 'aqīqah may be delayed until one becomes financially capable of performing it.
And Allah Almighty knows best.
What is the ruling on congregational prayer performed outside the mosque, and is it a condition that it be performed in a mosque in order to attain the full reward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is not a condition for attaining the reward of congregational prayer that it be performed in a mosque; however, a Muslim should still be keen to perform his prayers in the mosque, so as not to miss out on immense reward. This is due to the virtue of the mosque and of spending time there (i‘tikāf), the typically larger number of worshippers present, and the additional reward gained through walking to and from the mosque. On the authority of Abū Hurayrah (raḍiyallāhu ‘anh), the Messenger of Allah ﷺ said: "Whoever purifies himself in his house, then walks to one of the houses of Allah to fulfill one of the obligations Allah has prescribed, one of his steps will erase a sin, and the other will raise him a degree [in status]." (Related by Muslim)
Whoever misses congregational prayer at the mosque should still strive to pray in congregation at his workplace, or together with the members of his household. And Allah, the Exalted, knows best.
Is it permissible for a woman to uncover her face while performing Umrah (minor Hajj)?
The woman while in a state of Ihram (ritual consecration) is obligated to uncover her face and hands, but at the same time permitted to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.
When is it Sunnah to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered on the seventh day from the birth of the newborn. According to the sounder position, the day of birth itself is counted as the first of the seven days. Thus, for example, if the child is born on a Saturday, the 'aqīqah is to be slaughtered on the following Friday. If the child is born at night, the count begins from the day that follows. And Allah Almighty knows best.