Is it permissible for my son to wipe himself with tissue only at school, and does ritual impurity spread through sweat, and is the original state of things purity?

Using tissues for Istijmar (cleaning oneself after relieving nature) is sufficient to remove urine and feces if done with three purifying wipes. If the child knows the prescribed method of Istijmar, then the purification of the area from urine and feces is valid. As for sweat, it is excused and does not affect purity. Doubt about the transfer of impurity is not to be taken into account, because the default state is purity.