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 having an intention (Niyyah) for every prayer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Intention (Niyyah) is a pillar (Rukn) without which the prayer is not valid. The worshiper must have an intention for every prayer, meaning they must consciously intend the act of worship they are performing. Its timing must coincide with the opening Takbir (Takbirat al-Ihram). It is not a requirement to utter it verbally; rather, doing so is considered a recommended Sunnah. There are three levels of intention:
1-If the prayer is obligatory (Fard): It is mandatory to include the Intent (to pray), the Specification (which prayer, e.g., 'Asr), and the Obligation (recognizing it as a Fard). For example, one should bring to mind or say: 'I intend to pray the Fard of 'Asr.'
2-If it is a voluntary prayer restricted by a specific time or cause (Sunnah Muqayyadah): It is mandatory to include the Intent and the Specification. For example: 'I intend to pray the Sunnah before Zuhr' or 'I intend to pray Duha.'
3-If it is an absolute voluntary prayer (Nafl Mutlaq): It is sufficient to simply have the Intent to pray. For example: 'I intend to pray.'
And Allah the Exalted knows best.
Is it permissible for women to attend Tarawih prayer in the mosque?
Yes, it is permissible for women to attend Tarawih prayer in the mosque, provided they observe modesty, proper covering, and avoid mixing with men.
The Prophet ﷺ said: "Do not prevent the female servants of Allah from the mosques of Allah." [Bukhari and Muslim]
How is night prayer offered?
Night prayer is offering voluntary prayer after Maghrib and before Fajr (Dawn). As for Tahajjud, it is offering voluntary prayer at night after waking up voluntarily, and for Allah`s sake as He Says (What means): "And pray in the small watches of the morning: (it would be) an additional prayer (or spiritual profit) for thee: soon will thy Lord raise thee to a Station of Praise and Glory!" [Al-Isra`/79]. Offering Tahajjud is better than offering voluntary prayer before going to bed.