Do hands become defiled by touching everyday objects wet with sweat or water? Is walking barefoot on a spot wet with shoe residue considered defilement? Is it permissible to pray on a rug after it has dried in this case?

The fundamental principle regarding objects is their purity, and one should not deviate from this principle except with certainty. If a hand or any other object is impure, its impurity does not transfer unless the physical impurity itself remains on it. If the physical impurity is removed but its ruling (hukm) remains, then whatever touches it does not become impure. Slippers used in bathrooms or ablution areas often have wetness clinging to them from the remnants of ablution, so they do not make carpets impure. If the slippers are indeed impure, then the carpet becomes impure. However, if the carpet dries and the physical impurity no longer remains, then whoever steps on it will not have their feet become impure, even if they are wet. Excessive scrutiny in these matters can lead to obsessive doubts (waswasah).