What is the ruling on and method for making up missed prayers, according to the order and sequence, especially when combining making up prayers missed due to an excuse (awaiting purity after menstruation) and prayers missed without an excuse, taking into consideration making up two days' worth of prayers each day?

As long as you believe in the necessity of sequencing missed prayers, the correct order is to pray Dhuhr first, then Asr, then Maghrib, and so on. You may adopt the view of the Malikis, which permits making up two days' worth of prayers in one day. You can make up the prayers for these two days during your free time, starting with Dhuhr, then Asr, and so on.

You must promptly perform ghusl (definition) and pray as soon as you see purity. If you do so, you will not have any missed prayers to make up. You may also adopt the view of those who do not require sequencing between prayers, and thus you would make up your previous prayers. This opinion has strength and is more lenient for you.