Can someone break a make up fast (qada) after beginning it?
Once a person begins a make up fast (qada), it is prohibited to break it.
If they break it without a valid excuse, they are sinful, and the missed Ramadan fast remains a debt upon them.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah 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."
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.