What is the ruling on istinja' after urination, and is it done with water and soap or with water only?
Istinja' from urine and stool is obligatory. It is permissible to perform istinja' with water alone, or with toilet paper alone. The best way is with toilet paper then water. If one wishes to suffice with one of them, then water is better. And Allah the Almighty knows best.
What is the ruling on the ablution of one who touches women prohibited to him by a temporary prohibition (mahram bi-hurma mu'aqqata)?
It is prohibited to touch a woman who is temporarily prohibited (meaning one whom it becomes permissible to marry after the impediment is removed, such as the wife of a brother or paternal uncle). Touching her without a barrier invalidates ablution. And Allah the Almighty 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 it permissible to participate in an Udhiyah?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Regarding the sacrificial offering (Udhiyah), if it is a sheep or a goat, it only avails for one person. However, if it is a cow or a camel, it avails for seven people. It is permissible for the seven participants to have different intentions; for example, some may intend the sacrifice (Udhiyah), some the ’Aqiqah (sacrifice for a newborn), and others may simply intend to obtain meat.
It is stated in Hashiyat Qalyubi ‘ala Sharh al-Mahalli (Vol.4/P.256) and in Bushra al-Karim (p. 706):
"If one slaughters a camel or a cow on behalf of seven... it is permissible. Likewise, it is permissible if some of them intend their seventh for meat, and others intend their seventh for ’Aqiqah." And Allah the Almighty knows best.