Is it permissible to wipe the hand with a tissue if it touches an impurity and water or dust is not available, and does the hand become impure again if it sweats or touches something wet after being dried? Is it permissible to check the impurity of a wet spot on the carpet, and how can one ascertain the impurity of wet ground?

Impurity on the hand or similar surfaces is not removed except by washing it with water and using an ample amount; mere wiping is not sufficient. So, whoever does not wash impurity from their hand, their hand remains impure. As for the transfer of impurity from the hand to something moist or wet that it touches, there is a difference of opinion on this matter. Certainty dictates that everything is pure, so if you doubt the impurity of something, act upon the original principle, which is purity, and do not rule something impure except with absolute certainty.