What is the ruling of Islamic Law on supplicating with other than the transmitted (Ma'thur) supplications in prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible to supplicate (make du'a) during the prayer for anything from the affairs of religion or worldly life, and the prayer is not invalidated by doing so.
Shaykh al-Islam, the Imam an-Nawawi, may Allah have mercy on him, says: "Our madhhab [i.e., the Shafi'i madhhab] holds that it is permissible for a person to supplicate during it [the prayer] with anything that is permissible to supplicate with outside the prayer, from the affairs of religion or worldly life. He may say: 'O Allah, grant me wholesome sustenance, offspring, a house, and a beautiful young wife' — describing her; or: 'O Allah, free so-and-so from prison,' or 'destroy so-and-so,' and other such things — and none of this invalidates his prayer according to us. And this is also the view of Malik, ath-Thawri, Abu Thawr, and Ishaq." [Al-Majmu', Vol.3/P.454]. And Allah, the Exalted, knows best.
What is the ruling of Islamic Law on continuity (muwalah) in the ritual bath?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
Continuity (muwālāh) means washing each part of the body immediately after the previous one, without leaving a gap long enough for the first part to dry before the second is washed. According to the Shāfiʿī school, continuity is a recommended act (sunnah) in both the ritual bath (ghusl) and ablution (wuḍūʾ) — it is not obligatory.
It is stated in Nihāyat al-Muḥtāj ilā Sharḥ al-Minhāj: "Likewise, continuity in the ritual bath is recommended, just as it is in ablution."
Accordingly, continuity in the ritual bath is a sunnah. And Allah the Almighty knows best.
Does the time of Fajr (dawn) prayer start after the first Athaan (call for prayer), or the second one?
All perfect praise be to Allah,The Lord of The Worlds The time of Fajr prayer starts after the second Adhan. And Allah Knows Best.
Is the multiplication of reward for prayer in the sanctuaries of Makkah and Madīnah limited to the obligatory prayer, or does it include the voluntary prayer as well?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is no difference between the obligatory (farḍ) and the voluntary (nafl) prayer in the ḥadīth on the multiplication of reward in the two Sanctuaries (al-Ḥaramayn), because the authentic ḥadīths are unrestricted and general in wording. It is therefore not permissible to restrict or specify them without evidence. Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said: “Our school holds that the superiority of prayer in the two mosques of Makkah and Madīnah is not limited to the obligatory prayer, but includes both the obligatory and the voluntary.” [al-Majmūʿ Sharḥ al-Muhadhdhab, Vol.8/p.476]
The multiplication of reward applies to whoever prays in al-Masjid al-Ḥarām or anywhere within the sanctuary (ḥaram). Al-Shirwānī (may Allah have mercy on him) said: “The more correct view according to al-Nawawī is that the multiplication of the prayer’s reward covers the whole sanctuary, and is not limited to the mosque, nor to Makkah itself.” [Tuḥfat al-Muḥtāj with the glosses of al-Shirwānī and al-ʿAbbādī, Vol.10/p.95]. And Allah Almighty knows best.