Are the ideas indicating the necessity of making up all past missed prayers—whose number is unknown—correct for current prayers to be valid, and should we stop making up currently known missed prayers until all unknown old missed prayers are made up, or is it better to promptly make up any prayer missed currently due to an excuse?

A fard (obligatory) prayer that is missed due to an excuse (such as forgetfulness or sleep) must be made up as soon as one is able to, by consensus. As for a prayer that is abandoned without an excuse, there is a difference of opinion among scholars regarding the obligation to make it up, but most of them hold that making it up is obligatory. One must hasten to make up a prayer missed due to an excuse, and older missed prayers should be made up according to one's capacity. It is recommended to disregard obsessive thoughts (waswasah).