Is it valid to wash a prayer rug (sajadah) that has been defiled by a baby girl's urine by pouring water over it and drying it seven times without wringing it out? And does the ground underneath it become pure by the same method, given the presence of tiles, taking into consideration the Hanbali school's condition of washing seven times, and the Hanafis' opinion regarding what cannot be wrung out or flat ground?

What you are doing by being strict with yourself and always taking the stricter opinion in every matter, especially in this regard, is an act of extremism and stubbornness. It should be abandoned because it invites obsessive thoughts and may lead to deviating from the rulings of the Sharia (definition).

Regarding a ritually impure prayer rug, it becomes pure simply by pouring water over it, and the water that separates from it after its purification is considered pure unless it has changed due to the impurity. As for the ground and what is permanently attached to it, it becomes pure by drying, according to the preponderant opinion, or by pouring water over the impure spot until it covers the impurity.

In summary: You must abandon extremism in religion. As an ordinary person, it is sufficient for you to follow one of the trustworthy scholars without obligating yourself to adopt the stricter opinion in every matter.