Can Gege in the Qing Dynasty walk around the Palace Square, just outside the Palace Museum's ministerial courtyard?

No, women in the palace are not allowed to go out and clean the doors. The Forbidden City is divided into two departments by the dry cleaning door, Zijin City. The three main halls outside are used for major ceremonies or court meetings. The things in the dry cleaning door belong to concubines and minor children, and the things in the five halls belong to adult children. Under normal circumstances, Gege can't leave the inner palace. You can't wander around in your daily life unless major events are approved.