A problem in NP can have a (positive) solution verified in polynomial time. That's it. Requiring more than polynomial time to solve isn't part of the definition, and in fact it's an open question whether any problems in NP require more than polynomial time to solve.
Every single problem in P is in NP. What is believed but unproven is that some problems in NP are not in P.
Every single problem in P is in NP. What is believed but unproven is that some problems in NP are not in P.