Must the questioner make up for days she intended to make up, due to her belief that her fast was invalidated by looking at a young man, or by sitting in a backbiting gathering, or by shaking hands with her cousin? And are her fast and prayer valid on a make-up day when she felt a discharge, which might be maniy, after a conversation about desire? And must she make up the prayers if the discharge was maniy and she cannot remember when she performed ghusl?

Your fasting is valid, and you are not required to make up the day, as long as you completed the fast and did not break it. Your decision to make up the day is not considered an intention to nullify the fast. The discharge you felt does not invalidate the fast, whether it was manī or madhī, and ghusl (ritual bath) is not required unless you are certain that manī was discharged. We advise you to fear Allah and avoid what is forbidden, for fasting was legislated to attain taqwa (God-consciousness). Engaging in forbidden acts like backbiting does not invalidate the fast, even if it is contrary to taqwa.