Does passing wind from the anus affect ritual purity?
Ablution is nullified by the passing of wind from the anus, but washing the anus is not required as wind leaves no impurity.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is supplicating in Qunoot, at times of affliction, during obligatory and voluntary prayer a Sunnah, and should it be done before Rukoo` (bowing), or after it?
At times of affliction, it is a Sunnah that Muslims supplicate in Qunoot after the final Rukoo` of each obligatory, or voluntary prayer as individuals, or in congregation.
Does offering Udhiyah (Offering a Sacrificial animal at Aid Al-Adha) avails from offering Aqiqa?
Offering an Udhiyah doesn't avail a person from offering Aqiqa because each of which is a different act of worship. And Allah Knows Best.