What should one do if old and new missed prayers accumulate, and how should they be made up while observing their proper order?

You must preserve prayer at its appointed time, repent to Allah Almighty, and make up for missed prayers. If you do not know their number, then continue making them up until you are reasonably certain that you have fulfilled your obligation. The order between missed prayers is not obligatory, and one may adopt this view for ease. As for a missed Maghrib prayer, it must be made up before the 'Isha prayer. You must keep track of the number of missed prayers, even if it is only by writing them down. As for whispers of disbelief, their cure is to turn away from them, ignore them, and not pay them any attention.