Is everything a person walks on while wet considered ritually impure (najis) in the general sense? And did the prayer mat and the clothes on top of it become impure? And did the clothes on top of the mat become impure after stepping on them with a wet foot?

The person in question suffers from many obsessive thoughts. He should be advised to ignore them and not pay attention to them. The basic principle regarding things is purity (taharah), so one should not investigate the purity or impurity of something unless there is absolute certainty. What the brother walked on, whether it's the house floor or the carpets, is not considered impure. Similarly, prayer rugs do not become impure due to the possibility of him having stepped on them.