Questions about English expression Alltag von überall

I found that English users sometimes say "nope" when expressing negative intention. What is the difference between nope and no?
I originally guessed that /p/ was just the sound of airflow when people shut up. It has no practical significance. But I later found that people often use this word when they interrupt others. So it seems that there should be a big difference between this word and no.
I hope English users can help me solve this problem. Thank you!!!

Well, similarly, is the relationship between nope and no the same as that between yes and yep?

Well, it seems that you sometimes use nah to express negation Is this the same as no and nope?

From what I know, the only difference is that "nope/nah" and "yep/yup" are informal to very informal. They are mostly used when speaking, or on internet,but not for a mail or in other formal situations.

Sometimes, it might be to be a little nicer than just sayingg "no" because often "nope" sounds kind of softer.