What is the ruling on my prayer and what is the ruling on the mosque's prayer rugs if dry impurity got onto the car seat, and then I sat on it while I was very wet after performing istinja (washing after urinating or defecating) and entered the mosque, and how do I purify the car seat?

If the questioner is afflicted by waswas (obsessive whisperings) and doubts, he should ignore them, and the garment is pure; for the default status of things is purity until their impurity is proven.

If impurity is certain, then if the car seat is impure and you sit on it while your garment is wet, the wet part of your garment and whatever part of your body it touches becomes impure, according to the majority of scholars. If the wet garment becomes impure, whatever part of your body it touches also becomes impure, just as the spot you sat on in the mosque’s carpet becomes impure if it is affected by moisture that can transfer, and you must purify it. It does not become impure if you sit on it after your garment has dried.

Your prayer with an impure garment or body is invalid, unless you were forgetful. Scholars differ on its validity in that case, and Shaykh al-Islam Ibn Taymiyyah preferred the opinion of validity. However, if you repeat it, it is more cautious.

To purify the car seat, pour water over the impure spot until the water separates, unchanged by the impurity. If it is difficult to remove its color or smell, then these are excused.