What should be done if a person suspects a place is impure, and it is indeed impure? And is a place judged impure if a person whose trousers are impure sits there, and it is not known whether he perspired in it or not? And how does one differentiate between certainty and doubt regarding impurity?

Doubt and certainty do not coexist. Doubt about the impurity of a place means a lack of certainty, and certainty about its impurity means an absence of doubt. Speaking of a place being impure without certainty is illogical, and this vacillation between doubt and certainty is obsessive whispering (waswasah). The default state of things is purity. So, if you are not certain about the impurity of a place, then it is pure, and do not yield to obsessive whispering.