What is the ruling on a young woman going out to the mosque to perform Tahajjud prayer alone and returning around 3:30 AM, and does Tarawih prayer suffice for Tahajjud prayer, knowing that her going out alone causes her mother anxiety?

It is better for a woman to pray in her home. The more secluded and farther from mixing with men her prayer place is, the better. This applies equally to both obligatory and supererogatory prayers. It is permissible for a girl to go out for Tahajjud prayer, wearing the legal (Shar'i) hijab, if the path is safe and with her parents' permission, and she will receive reward for it. However, her praying in her home is preferable, especially given her parents' concern.