If $P = NP$ would this imply that polynomial time reduction from an $NP$- to a $P$-problem would be possible? And if $P\neq NP$ does it imply that a polynomial time reduction from an $NP$- to a $P$-problem would be impossible?
Since $P \subseteq NP$, every problem in $P$ is also in $NP$. So in both cases there is an $NP$-problem that can be reduced to a $P$-problem. Simply choose a problem in $P$ as the $NP$-problem and the same problem as the $P$-problem.
If you replace $NP$-problem by $NP$-complete problem, both of your statements hold.