Is Fajr (dawn) Sunnah a confirmed one?
All praise be to Allah The Lord of The Worlds Of course, it is a confirmed Sunna which the Prophet (PBUH) used to offer in a regular basis. And Allah Knows Best.
What is the ruling on one who doubts washing a limb before or after finishing ablution?
If a person performing ablution doubts leaving the washing of a limb from the limbs of ablution during his ablution, he must repeat washing that limb and wash what comes after it. If he doubts leaving the washing of a limb after finishing the ablution, there is nothing upon him. And Allah the Almighty knows best.
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.