Let $A$ be a representation-finite block of a schur algebra with $n \geq 2$ simple modules. Then the global and dominant dimension of $A$ are equal to $g=2n-2$. You can find quiver and relations for example in theorem 4.1. of https://www.sciencedirect.com/science/article/pii/S0022404918300380 when all lambdas are equal to one. For $n=2$ this is just the Nakayama algebra with Kupisch series [2,3].

Computer experiments suggest that we have $A \cong \Omega^g(D(A))$ as $A$-bimodules (while the projective dimension of $D(A)$ should be equal to $2g$).

>Question 1: Is there an easy argument for this, avoiding heavy computation?
>
>Question 2: Is there a deeper reason for such an isomorphism in case it is true, does it hold for a more general class of algebras?

My guess is that for certain algebras we have that $\Omega^i(A) \cong \Omega^j(D(A))$ for some i,j which would have interesting consequences for Hochschild homology and cohomology. But I was not able to think of any concrete conditions despite finding some examples. 

Another class of examples seem to be the Auslander algebras of $K[x]/(x^n)$, where $A \cong \Omega^2(D(A))$.