If you pray Tarawih in one mosque, then you find another mosque where prayers are still going on, and you go and pray with the people, there is nothing wrong with you doing that, but you should not pray Witr twice.

A woman may pray Tarawih at home according to whatever she is able to do, paying attention to the Sunnah as much as possible. For more, please see the detailed answer.

Uttering the intention out loud when one is going to pray is bid`ah whether that is for Tarawih prayers or any other prayer.

It is prescribed for the Muslim to perform Tarawih prayer after ‘Isha on the first night of Ramadan, which is the night on which the new moon is sighted or the Muslims complete thirty days of Sha`ban.