What is the ruling on a traveling woman praying in a train while seated due to the presence of men, and should she shorten her prayers at her family's home or at her place of study, and what is the ruling on shortening prayers if the duration of the journey is only three days?

A traveler must pray in their current state if they fear the time for prayer will expire, even if they cannot perform all of its pillars. So, if the time for Dhuhr and Asr prayers arrives and you are on a train that continues to move, you must pray them on the train. If you are able to pray standing, then it is not permissible for you to pray sitting. As for Maghrib and Isha prayers, if it is possible to perform them on the train, fulfilling all their pillars and conditions, then pray them. Otherwise, delay them until the train arrives at its destination so you can perform them fulfilling all their pillars. The country where you intend to reside permanently is considered your homeland, and you should not shorten prayers there. As for the other country where you do not intend permanent residence, it is permissible to shorten prayers if the intention of stay is four days or less; otherwise, according to the majority of scholars, prayers must be prayed in full.