I used artificial nails on my wedding night, but I was surprised to find out that they require at least a week to remove. However, I performed all my prayers during that period. What is the ruling on those prayers?

 

If these nails prevent water from reaching one's skin then your prayer are invalid and obliged to redo all of them. And Allah 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 offer mandatory prayer while sitting?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.                                                                                                                                                                                                                            Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.

What is the ruling on a vowed animal sacrifice?

A vowed animal sacrifice is to be distributed amongst the poor and needy, and neither the vow-maker, nor those supported by him are to eat from it.