What is the ruling regarding our prayer when the sitting areas and a sister's clothes are contaminated with dry urine due to involuntary urination, and is the dryness of urine considered a purifier that removes impurity from the place and clothes?

There is no need for worry, as this is a form of extremism and obsessive doubt (waswasa). You should deal with carpets and furniture normally, and nothing is ruled ritually impure (najis) except with absolute certainty. If the impure object is dry, the impurity does not transfer to dry objects it touches. However, if one of them is wet, there is a difference of opinion regarding the transfer of impurity. If you are afflicted with waswasa, you are permitted to take the easier opinion and act upon it.