Which hour coincides with the last third of the night?
This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.
What is the ruling on performing the Witr prayer as a single rak`ah?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to perform the Witr prayer as a single unit (rak‘ah). It was narrated from Ibn ‘Umar that a man asked the Messenger of Allah ﷺ about the night prayer, and the Messenger of Allah ﷺ replied: 'The night prayer is offered two by two (mathna mathna). If one of you fears the approach of dawn, let him pray a single rak‘ah to make what he has prayed odd-numbered (Witr) for him.' (Related by al-Bukhari & Muslim)). However, limiting the prayer to only one rak‘ah is considered 'contrary to the preferred way' (Khilaf al-Awla).
It is stated in Al-Minhaj al-Qawim Sharh al-Muqaddimah al-Hadramiyyah (p. 137): 'The minimum of Witr is one rak‘ah, but limiting it to that is contrary to what is best.'
The most complete form of Witr is eleven units, while the minimum level of 'perfection' is three units. It is stated in ‘Umdat al-Salik (p. 60): 'The minimum of Witr is one rak‘ah, and its maximum is eleven, performing the taslim (salutation) after every two units. The lowest level of perfection is three units with two separate taslims (meaning 2+1).' And Allah the Exalted knows best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible for a woman to perform the obligatory prayer immediately after the adhan and before the Iqamah is performed in the mosque?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Yes, it is permissible for a woman to perform the obligatory (Fard) prayer immediately after the Adhan (call to prayer) and before the Iqamah (call to commence prayer) is performed in the mosque. And Allah the Exalted knows best.