What is the ruling regarding a bed and prayer area after wiping them with a towel dampened with water and Lysol, knowing that impurity is removed by pouring water? And are objects touched by a hand, after it has touched the anus and been purified with hand sanitizer, considered impure?

The impurity of a mattress is not ruled by mere doubt, as the original state is purity. Nothing is required of you as long as you are not certain about the impurity of the syringe and the mattress. As for your hand, if you are certain of its impurity, then washing it is obligatory. If you are not certain, then nothing is required of you. If you leave it unwashed when it is obligatory, some scholars hold that impurity does not transfer from it to whatever it touches, even if it is wet. It is permissible to act upon this opinion for those who suffer from obsessive whispers (waswas). It is advised to ignore these whispers and turn away from them.