What is the Du`a (supplication) of Istikhara (guidance prayer)?
O Allah, I consult You as You are All-Knowing and I appeal to You to give me power as You are Omnipotent, I ask You for Your great favor, for You have power and I do not, and You know all of the hidden matters. O Allah! If you know that this matter (then he should mention it) is good for me in my religion, my livelihood, and for my life in the Hereafter, or he said: "for my present and future life" then make it (easy) for me. And if you know that this matter is not good for me in my religion, my livelihood and my life in the Hereafter, or he said: "for my present and future life" then keep it away from me and take me away from it and choose what is good for me wherever it is and please me with it."
What is the meaning of the Prophetic statement that a boy is held in pledge (murtahan) for his 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The first interpretation: That if the boy dies in infancy without an 'aqīqah having been performed on his behalf, he will not intercede for his parents on the Day of Resurrection. This is the position of Imam Aḥmad ibn Ḥanbal, and Imam al-Khaṭṭābī concurred with him, stating: "The finest of what has been said regarding this matter is the position adopted by Aḥmad ibn Ḥanbal." — [Fatḥ al-Bārī by Ibn Ḥajar,{Vol.9/P.594]
The second interpretation: That the child is likened to a pledged object (marhūn) — one from which full benefit and enjoyment cannot be derived until it is redeemed. A blessing is only made complete upon the one blessed when they fulfil the obligation of gratitude (shukr), and the prescribed expression of gratitude for this particular blessing is what the Prophet ﷺ established as Sunnah — namely, the slaughtering of the 'aqīqah on behalf of the newborn as an act of thankfulness to Allah the Almighty and as a supplication for the wellbeing and safety of the child. This is the position of Mullā 'Alī al-Qārī. See: [Mirqāt al-Mafātīḥ Sharḥ Mishkāt al-Maṣābīḥ, Vol.7/P.2688]
And Allah Almighty knows best.
Is it permissible to slaughter a sheep on Eid al-Adha with the intention of charity (sadaqah) on behalf of a sick person, and with the intention of udhiyah (sacrificial animal) at the same time?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The udhiyah (sacrificial animal) is an act of worship intended for its own sake; therefore, it must be slaughtered with the intention (niyyah) of udhiyah.
Whoever intends to draw closer to Allah through this act of obedience with the intention of udhiyah, and also intends to gift the reward to another person, there is no Islamic objection to that. And Allah Almighty knows best.
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.