Can I complete fasting and prayer after the black discharge and yellow discharge, knowing that I stopped fasting and praying on the two previous days, and can I continue taking the period-delaying pills and purify myself for 'Umrah if bleeding occurs again?

If the menstrual bleeding completely ceases and the woman sees purity, then she is pure and should perform ghusl (definition), pray, and fast. If the blood returns after that, it is considered menstruation, which prohibits prayer and fasting. It is permissible for a woman to take menstruation-stopping pills if they do not cause harm, but purity is achieved by the cessation of blood, not merely by taking the pills. If the blood does not cease, then she is menstruating, and she is permitted to enter into ihram (definition) for 'Umrah (definition) while menstruating, but she should not circumambulate the Kaaba until she becomes pure.