Is the defilement of a worshiper's feet, when walking on a mat that may have been touched by an impure dog, considered an impurity, and does the presence of restrooms inside the mosque cause a problem for prayer?

The default state of a mat is purity, and merely the dog lying on it does not render it impure, because its fur is pure. Even if its fur were impure, what it touches would not be judged impure unless one of them was wet. So, if you do not know that the dog's body or the mat was wet at the time the dog lay on it, the default state of the mat is purity.

Dogs used to enter the mosque during the time of the Prophet, peace be upon him, and the Companions did not sprinkle anything. Your doubts about the transfer of saliva do not negate the original purity of the mat.

Furthermore, the contiguity of the restroom to the mosque and opening its door inside it does not affect the validity of prayer or the purity of the mosque's carpet.

We advise you to stop excessive doubt (waswasa) and to act upon the principle that the default state of things is purity unless their impurity is certain, as Sheikh Al-Sa'di said: "And the default state of our things is purity And the earth, clothes, and stones."