Must I make up for all the prayers I performed with an incomplete ablution, and is it permissible to adopt the view that allows for making them up at one's leisure, or must they be made up immediately? And how can they be made up immediately if there isn't enough time?

If you unintentionally omit a part of wudu (ablution) that is obligatory to wash, there is a difference of opinion among scholars regarding the necessity of making up (qada’) the prayers. Sheikh al-Islam (Ibn Taymiyyah) holds that it is not obligatory, while the majority (jumhur (definition)) consider it obligatory. Making up the prayers is immediately obligatory, provided it does not harm your body or livelihood. If making up the prayers affects your studies, then make up what is easy for you without harm. You may adopt the view of the Malikis, which allows making up two days' worth of prayers with each day [of current prayers]. It is not obligatory to make them up consecutively; rather, it is permissible to distribute the missed prayers throughout parts of the day, while observing the correct order.