I search for noetherian commutative rings having distinct prime ideals $P⊂Q$ with no primes between them, and where $\operatorname{grade}(Q)≠\operatorname{grade}(P) +1$, or $\operatorname{height}(Q)≠\operatorname{height}(P)+1$.
If $R$ is Cohen-Macaulay, are the left and right sides of the above expressions equal?
Thanks for cooperation!
Let $R=K[x, y]/(x^2, xy)$, $P=(x)$, and $Q=(x, y)$. Both are prime ideals, there is no prime between them, and both have grade zero.
For the height part of your question consider Nagata's example of a noetherian local domain which is not catenary and has dimension $3$. It contains a maximal chain of prime ideals $(0)\subset\mathfrak p\subset\mathfrak m$, and $3=\operatorname{ht}\mathfrak m>\operatorname{ht}\mathfrak p+1=2$.
Such a phenomenon can't occur in a Cohen-Macaulay ring: localize at $Q$ and recall that a Cohen-Macaulay ring is catenary.