Have you seen the evidence going around that Papyrus knows a lot more than he let's on? He remembers Frisk after a True Reset, predicts many events that take place later in the game (Mettaton Ex, what happens when you reach Asgore, what outfit you wear, Asgore trimming a hedge in the shape of his face), and he seems to have the same powers as Sans, even if he rarely shows them. What do you think?
We have not seen this evidence. However, Papyrus shouldn’t remember Frisk after a True Reset, considering the purpose of the True Reset is to completely erase everyone’s memories. It’s actually after regular resets that Papyrus has the feeling of déjà vu about Frisk!
In regards to how Papyrus seems to predict events that happen later on in the game, there is a particular flower that likes to give Papyrus advice and predictions – Flowey!
In regards to strength, it is definitely hinted that Papyrus is stronger than he lets on. He likely has the same gasterblaster attack Sans does, considering his special attack would have “BLASTED” you.
Plus, Undyne herself vouches Papyrus’ strength.
However, it’s unlikely that Papyrus is all knowing. Considering what Papyrus has said to Sans, it’s known that Flowey is feeding Papyrus information, likely from his experiences in other timelines he had run through before Frisk fell down. Although there is no denying that Papyrus also has a habit of leaning on the fourth wall on occasion, like his brother.