What is the ruling of Islamic law regarding leaving some traces of writing or stains on the Mus'haf, and its pages crumbling while attempting to remove them, and what is the ruling on placing Mus'hafs and religious papers in the same cupboard as clothes?

It is not necessary to remove traces and stains from the Mus'haf if that is difficult. The default state of things is purity. What has separated from the pages of the Mus'haf does not take the ruling of the Mus'haf unless it contains mention of Allah. There is no harm in placing what contains mention of Allah on a shelf, as long as it does not involve desecration.