Is hair extension permissible?

All Perfect Praise is due to Allah, The Lord of The Worlds 

It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.

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."

Is it permissible to give Zakah (obligatory charity) to one`s poor sister, or brother?

It is permissible for one to give the Zakah to his poor sister, or brother if providing for them isn`t due on him, and the sister doesn`t receive sufficient provision from whomever that is due on. And Allah Knows Best.

Does passing wind, from the anus without a smell, invalidate prayer?

If a person is certain of having passed wind even if it was with no smell, then both his/her ablution and prayer are invalidated, thus he/she should remake ablution and re-perform prayer.