What should a woman do if yellowish discharge appears after the sign of purity and ghusl, and continues until the end of the menstrual period without another sign of purity appearing? And does she have to repeat ghusl? And what if the yellowish discharge continues until the end of the menstrual period without any sign of purity appearing at all; should she perform ghusl or not?

There is a difference of opinion among scholars regarding yellowish and brownish discharges. According to those who do not consider them to be menses at all, or only if they are connected to blood, then you should not pay attention to them. However, according to the opinion of the majority of scholars, they are considered menses as long as they occur within the possible period [of menstruation], and you do not become pure until they cease. The Hanbali madhhab (definition) considers them menses only during the period of one's regular habit, so anything beyond that is not menses. You may follow whichever scholar you wish, but the most cautious approach is to act upon the opinion of the majority.