Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
What is the ruling in Islam on castrating animals to use them for work?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
This isn`t permissible because jurists said: "It is permissible to castrate the animal if it is eaten and this procedure is done at early age." And Allah The Almighty Knows Best.
What is the ruling on smoking?
All Perfect Praise is due to Allah, The Lord of The Worlds
Smoking is forbidden (Haram) since it is harmful to one`s health as confirmed by trustful physicians, and Islam has forbidden taking anything that causes harm. The Prophet (PBUH) said: "There should be neither harm nor reciprocated harm." [Ibn Majah and Ahmad]. And Allah Knows Best.
Is it permissible to repeat the Istikhara (seeking guidance) prayer, and is it a condition that one should sleep after offering it?
All perfect praise be to Allah,The Lord of The Worlds Istikhara is offered once, but it is permissible to repeat it, and sleeping after offering it is not a condition for its validity. And Allah Knows Best.