Does a woman receive the same reward as a man and the reward for congregational prayer when she goes to the mosque to perform Tarawih prayers or any obligatory prayer, while adhering to full, شرعي hijab, or is this reward exclusive to men only?

A woman's prayer in her home, even if she prays alone, is better and carries a greater reward than her prayer in the mosque, even if it is in congregation. This is by the agreement of the jurists, because their prayer at home is superior, based on the marfu' hadith of Umm Salamah: "The best mosques for women are the depths of their houses." As for the virtue of congregational prayer in the mosque, it is specific to men, because they are commanded to go out for it, except for the Eid prayer, which also carries a multiplied reward for women because they are commanded to go out for it.