Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
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.
Which is better in night prayer: long standing (Qiyam) or long prostration (Sujood)?
Long standing (Qiyam) is better in prayer. However, a Muslim should also observe tranquility and composure in their prayer.
What should someone do if they fasted for only 28 days in their country and then traveled to a place where Eid has already been declared?
If a person fasts in their country and then travels to another country where Eid has been declared, they must celebrate Eid with the people of that country.
● If their total fasts add up to 29 days, nothing is required of them.
● However, if they have only fasted 28 days, they must make up one day after Eid, because an Islamic month cannot be only 28 days.