If a Muslim doubts the existence of something forbidden in an item he uses, or impurity in his place of prayer, and he is not certain: Is he obliged to ask and verify, or should he rely on the original presumption (purity or permissibility)? And is his doubt treated the same as ignorance or forgetfulness if the matter becomes clear later?

The established principle is that certainty is not removed by doubt, and acting upon it closes the door to obsessive whispers (waswas). So, if you doubt that something contains a forbidden substance, the default is that it does not contain it. If you doubt the defilement of a certain spot, the default is that it is not defiled. Build upon this default in all matters, and you are not required to ask or research. If it later becomes clear to you that the item was impure, your ruling is like that of one who prayed with impurity unknowingly, and the preponderant opinion is that he is not required to repeat the prayer. If the opposite of the default does not become clear, then the prayer is valid.