Is the prayer considered invalid if the worshipper intentionally rushes the final Tashahhud due to a feeling that there might be some impurity, and then it becomes clear to him that there is none?

The final Tashahhud is a pillar according to the Hanbalis and Shafi'is, and the prayer is invalidated by its omission. The minimum sufficient for the Tashahhud is: "All greetings are for Allah, peace be upon you, O Prophet, and the mercy of Allah, peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, and I bear witness that Muhammad is His servant and His Messenger," or "that Muhammad is the Messenger of Allah." Praying upon the Prophet in the final Tashahhud is obligatory according to the Shafi'is and the relied-upon view among the Hanbalis. Prolonging the Tashahhud is not recommended. Your prayer is valid if you performed what is sufficient of the Tashahhud and the prayer upon the Prophet, according to those who hold that prayer is invalidated by their omission. One must disregard the whisperings of the self.