What is the ruling on touching papers containing the Basmalah (In the name of Allah, the Most Gracious, the Most Merciful) with a stain suspected to be urine or egg white, and is it obligatory to wash these papers despite years having passed since the incident and the uncertainty of their existence or damage?

We ask Allah to cure you of whispers (waswas) and inspire you with guidance. Everything that contains the name of Allah must be revered and protected. Nothing becomes impure by doubt; the default state of the examination paper is purity unless you are certain of its impurity. Even if it were to become impure, a small amount of impurity is excused, and there is no sin upon you for adopting this view as long as you are afflicted with whispers. Disbelief has no entry here, as you did not intend disrespect or insult. It is not obligatory to purify the paper with water if it leads to its damage, and you are not required to search for it in the examination committees.