Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
Should a prayer performer who forgets to offer the middle Tashahod (saying ash-hadu anala ilaha ila-alah) offer Sujood As-Sahw (prostration of forgetfulness)?
He/she should offer two prostrations of forgetfulness at the end of the prayer. And Allah Knows Best.
What is the ruling on eating or drinking during the second Adhan?
It is not permissible to eat or drink during the second Adhan because it announces the break of dawn and the obligation to start fasting. Allah Almighty says {what means}:"and eat and drink until you can discern the white streak of dawn against the blackness of night." [Al-Baqarah/187].
Whoever eats or drinks during this time invalidates their fast, and they must refrain from eating for the rest of the day and make up for the missed fast later.
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.