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.
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.
Is it permissible to offer mandatory prayer while sitting?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.
What is the ruling of Islamic Law on sitting between the two prostrations, and what is the ruling on the dhikr recited during it?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
Sitting between the two prostrations (al-jalsah bayna al-sajdatayn) is one of the pillars (arkān) of the prayer, while the dhikr recited during it is Sunnah according to the Shāfi'ī scholars. It is recommended (mustahabb) to say what was narrated from Ibn 'Abbās (may Allah be pleased with him), who said: "The Messenger of Allah (peace and blessings be upon him) used to say between the two prostrations in the night prayer: 'Rabbi ighfir lī, warhamnī, wajburnī, warzuqnī, warfa'nī' (My Lord, forgive me, have mercy on me, make good my deficiencies, provide for me, and raise me)." If one were to say a different supplication instead, such as "Rabbi ighfir lī" (My Lord, forgive me), the prayer would still be valid.
If the worshipper omits this dhikr between the two prostrations, whether intentionally or out of forgetfulness, their prayer remains valid and nothing is required of them, though they miss the reward of following the Sunnah. Leaving it out does not necessitate the prostration of forgetfulness (sujūd al-sahw).
It is stated in Hāshiyat al-Bājūrī 'alā Sharḥ Ibn Qāsim 'alā Matn Abī Shujā' (Vol.1/P.298): "The eleventh — i.e., of the eighteen pillars — his statement: 'sitting between the two prostrations,' meaning even in a supererogatory (nafl) prayer... his statement: 'with the supplication that has been narrated concerning it,' meaning: 'Rabbi ighfir lī, warhamnī, wajburnī, warfa'nī, warzuqnī, wahdinī, wa 'āfinī' (My Lord, forgive me, have mercy on me, make good my deficiencies, raise me, provide for me, guide me, and grant me well-being). Al-Ghazālī added: 'wa'fu 'annī' (and pardon me). Al-Mutawallī also added: 'Rabbi hab lī qalban taqiyyan, naqiyyan min al-shirk, bariyyan lā kāfiran wa lā shaqiyyan' (My Lord, grant me a heart that is God-fearing, pure from associating partners with You, innocent, neither disbelieving nor wretched)."And Allah, the Most High, knows best.