Is a prayer rug considered impure after drops of water from a wet tissue fall onto the floor, and do other things that touched that prayer rug become impure, and is the water that fell from the tissue pure or impure?

This question indicates some degree of waswasah (whisperings/obsessive thoughts). We advise the questioner to disregard these whisperings and not pay attention to them. As long as the water that has fallen is doubtful in its impurity, the default is that it is pure. Impurity is not determined merely by doubt, and everything touched by this water is considered pure.