Are the socks, clothes, and mosque carpets defiled if they are touched by pizza grease that was transferred by a knife that cut a pork pizza, and is it obligatory to repeat the prayer?

The jurists have agreed upon the impurity of pork due to the Almighty's saying: "For it is defilement (Rijs)." If a knife is used to cut pork and then used to cut other food without being washed, the part of the other food that came into contact with the knife becomes impure and must be removed. If the knife was wiped, then there is hope that there is no harm. As for the impurity of mosque carpets and clothes, the default state of things is purity, and what you mentioned are mere whispers (waswas); nothing has been affected by the impurity of the pig because if any impurity were present, it would be minor and would disappear, just as was the case with dogs that would urinate in the mosque during the time of the Messenger of Allah, peace be upon him, and they did not sprinkle anything from that.