What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?
Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.
What is the ruling on someone who insults or deceives others while fasting?
Insulting or deceiving others is forbidden (haram) and diminishes the reward of fasting.
The Prophet ﷺ warned against deception, saying: "Whoever deceives is not one of us." [Narrated by Al-Tirmidhi]
However, their fast remains valid despite the sin.
If someone delays making up fasts for a year, can they give fidyah before performing qada?
Yes, it is permissible to give fidyah before making up the missed fasts (qada), because each is an independent obligation, and there is no required order between them.