What is the ruling on making up for the missed days of Ramadan due to menstruation, given that the number of days is not remembered, and is it necessary to wait a week after the menstrual period if it does not exceed five days, and what is the ruling on making up for the day on which one vomited involuntarily?

What you were told about needing to wait a week after your menstrual period ends is incorrect. Rather, you must perform ghusl (definition), pray, and fast immediately upon seeing one of the two signs of purity. You must make up the days you missed fasting in Ramadan (definition) before the next Ramadan arrives. If one delays making up the fasts until the next Ramadan begins, a minor expiation (feeding a poor person one mudd for each day) is obligatory if they knew the prohibition of delay. Otherwise, only making up the fast is required. You must make up the days about whose number you are uncertain, in order to clear your conscience. If you waited after your period ended during Ramadan, you must make up the prayers and fasts of those days. If you did so outside of Ramadan, you must make up the prayers only. You are also required to make up the day on which you vomited intentionally, or unintentionally but did not complete the fast of that day.