Must a person refrain from eating for the rest of the day if they break a fast of a vow (nadhr) or a make up fast (qada)?
Whoever observes a vowed fast (nadhr) or a makeup fast (qada) is prohibited from breaking it without a valid excuse. If they break it without a legitimate reason, they are sinful.
However, they are not required to refrain from eating for the rest of the day, because such restraint is only required out of respect for the month of Ramadan, not for other types of fasting.
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
Does using a wet miswak while fasting break the fast?
A fasting person should ensure that the miswak is dry when using it.
However, if the miswak is slightly moist but does not release any liquid when squeezed, then its use does not break the fast.
What is Tarawih prayer?
Tarawih prayer is the night prayer (Qiyam) performed during the month of Ramadan. It is called "Tarawih" because Muslims used to prolong the standing (qiyam) in it and take brief rests after every four rak‘ahs.