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 should the one performing the slaughter say at the time of slaughtering the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah to say the following supplication at the time of slaughtering the 'aqīqah:
"Bismillāhi wallāhu Akbar, Allāhumma laka wa ilayka, hādhihi 'aqīqatu fulān."
("In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of [the child's name].")
This is based on what al-Bayhaqī narrated in Al-Sunan al-Kubrā (Vol.9/P.511), that the Prophet ﷺ said: "Slaughter it in his name and say: In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of so-and-so."
And Allah Almighty knows best.
Is an animal that results from crossbreeding between two types of livestock (an'ām) valid [for sacrifice]?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
An animal that results from crossbreeding between two types of livestock (an'ām) is valid [for sacrifice], with consideration given to the higher of the two required ages. Thus, in the case of an animal crossbred between sheep (ḍa'n) and goats (ma'z), only one that has reached two years of age is valid [for sacrifice]. And Allah, the Most High, knows best.
Is it permissible for women to attend Tarawih prayer in the mosque?
Yes, it is permissible for women to attend Tarawih prayer in the mosque, provided they observe modesty, proper covering, and avoid mixing with men.
The Prophet ﷺ said: "Do not prevent the female servants of Allah from the mosques of Allah." [Bukhari and Muslim]