Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
Is it permissible for a person who is in a state of Janabah (Ritual impurity) to clip his nails, or to cut his hair?
It is permissible for those in a state of Janabah and menstruation to clip their nails and to shave whatever hair that is permissible to be shaved. And Allah Knows Best.
What is the ruling on paying a sum of money to Saudi citizen in return of having Saudi residence? It is worth noting that this transation is labelled as a "worker's visa" but I am not going to work as such only to facilitate having it?
Allah The Almighty Says (What means): "O you who have believed, fear Allah and be with the truthful" [At-Tawbah/119]. False statements contradict truthfulness, and it is not permissible to write them. A Muslim has the right to live in any Islamic country, and those who prevent him will be accountable before Allah about their action and the justification for the prohibition. And Allah Knows Best.
Is it permissible to offer Istikhara prayer (seeking guidance) after Witr Prayer (A confirmed Sunnah which is offered after Isha prayer and before Fajir prayer with a minimum of 3 Rak`as and a maximum of 11)?
It is permissible to offer Istikhara before and after Witr prayer as it is permissible at any time except that in which prayer is disliked. And Allah Knows Best.