What is the ruling on purifying a prayer mat and the place of prayer after doubting their impurity from the separated water of ablution, and is one required to purify the remaining places to which impurity is feared to have transferred?

Leave aside these whispers [waswasa], and pay no attention to them. As long as you are not certain that something has become impure, then the default state is purity. Nothing is judged to be impure merely based on doubt. If you are certain that impurity has transferred to a specific spot, then it must be purified for prayer to be performed on it. What you have mentioned is a result of obsessive whispers [waswasa], so strive against yourself to rid yourself of them and seek Allah's help.