What is the ruling on reciting the takbīr from Sūrat aḍ-Ḍuḥā to Sūrat an-Nās upon completing the recitation of the Qur'an (khatm)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is recommended to recite the takbīr from the beginning of Sūrat aḍ-Ḍuḥā through to the end of Sūrat an-Nās, and this is the recitation practice of the people of Makkah.
It is stated in Nihāyat az-Zayn (p. 65): "It is sunnah to begin each sūrah with the basmalah, since it constitutes a verse of it. It is also sunnah to say, before each sūrah — from aḍ-Ḍuḥā to the end of the Qur'an — 'Lā ilāha illallāh, wallāhu akbar' (There is no god but Allah, and Allah is the Greatest), based on what has been narrated regarding Imam ash-Shāfi‘ī (raḍiyallāhu ‘anh), who heard a man reciting this takbīr in prayer, and said to him: 'You have done well, and you have hit upon the sunnah.'"
And Allah, the Exalted, knows best.
Does collecting saliva and swallowing it break the fast?
Swallowing collected saliva does not break the fast, but it is an unnecessary and meaningless act.
Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?
Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.
Why was the exact date of Laylat al-Qadr concealed?
The wisdom behind this is to encourage Muslims to strive in worship throughout all the nights of Ramadan, or at least in the last ten nights. By doing so, they will earn the reward of observing Laylat al-Qadr as well as the reward of worshiping on other blessed nights.