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.
What should I do in order to wake up for Fajr (dawn) prayer?
All perfect praise be to Allah, The Lord of The Worlds What should I do in order to wake up for Fajr (dawn) prayer? Answer: Some of the acts that aid in offering Fajr prayer are: 1- Going to bed early. 2- Willingness to wake up. 3- Reciting Al-Korsi verse from Chapter Al-Baqarah. 4- Taking refuge in Allah from the cursed devil, and making ablution immediatley after being awake, but it is preferable to ask someone to help in this regard to perform the Fajr prayer. And Allah Knows Best.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.
What is the ruling on one who doubts washing a limb before or after finishing ablution?
If a person performing ablution doubts leaving the washing of a limb from the limbs of ablution during his ablution, he must repeat washing that limb and wash what comes after it. If he doubts leaving the washing of a limb after finishing the ablution, there is nothing upon him. And Allah the Almighty knows best.