Does papyrus actually have determination but very limited which explained why he able to survive from 20,000+ damage from player
Maybe certain monster have determination while the other (like the citizen of the underground) have little to none
Also according to undyne, papyrus is stated to be tough but innocent so maybe from all the training caused him to possess a determination?
Yes Monsters can have determination, but it seemed potentially undetectable until Alphys discovered it with the DT Machine that she made, I like to entertain the thought that there's a good reason that Undyne can change into the Undying and Alphys working with DT. I don't think Alphys experimented on her or anything, but something happened to Undyne to cause that. What's that? She's friends with Papyrus??? Don't worry about it. He would never do something evil ever.
Undyne calls Papyrus too innocent and nice to become a guard, she remarks that he would be ripped into little smiling shreds. But it's important to note that Toby writes every character like a person, and they don't know everything, like us, they only get a view at a character, and specifically a view that the character wants to give us.
Before her battle, Undyne is scary, and meant to be feared- she appears in armor, chases us without speaking a word, and attacks us in the dark.
But we run away, and give her water at hotland, and meet her to befriend her at Papyrus' insistence (Papyrus casually carrying us to the pacifist end).
Anyway, despite those around him, Papyrus is not too nice, naive, or innocent to be a royal guard. But if Undyne had said it properly: Papyrus is too nice because he won't kill anyone. He doesn't say it directly but he absolutely refuses to. He even avoids saying the word "Murder" properly, is he scared of killing that badly? Or is he avoiding it because he is speaking to a character that is a child (the player)?
Papyrus can control his damage to the point he can't take any, even when the player is on 1HP (tested). This is better than Toriel's control, as she can accidentally kill you- and modifies her battle to try and avoid that, but she can still take that last bit of dmg, and you get a fun horrified look from her if you get it right. Anyway Papyrus can't do that, if you modify the game so he takes more dmg, it just crashes, isn't that fun?
ALSO A FUN FACT: You can't dust Papyrus until after he spares you. So when he spares us in a murder run, he knows exactly what we could do, but he's letting us make that choice.
On the DT stuff:
Papyrus makes 2 sets of remarks that heavily imply he either had a save point, or still has them. His joke? about his brother pranking you across time and space. "I HATE IT WHEN HE DOES THAT!"
But we were pranked because we could load and save.. so while everyone uses this quote as a "PAPYRUS' BROTHER KNOWS ABOUT SAVE FILES???" It is Papyrus being the one that's getting trolled.. which means HE would have to be the one that loads and reloads and gets pranked.. HE would be the one that knows about the save files.