I do not know much on recent developments related to the first three questions asked. However, i know of some old results related mainly to the fourth question:
If $A_1$ is the Weyl algebra over an alg closed field of zero char, with the two generators denoted $p,q$ and $I$ is a non-zero, right ideal, then $M_2(End_{A_1}(I))\cong M_2(A_1)$ and $A_1$ is Morita equivalent to $End_{A_1}(I)$. Furthermore, these algebras are not generally isomorphic: Pick for example $I=p^2A_1+(pq+1)A_1$. Its endomorphism ring is isomorphic to $\{x\in Q|xI\subseteq I\}$, where $Q$ is the quotient division ring of $A_1$. This is not isomorphic to $A_1$ but it is Morita equivalent to it. If you are interested in this example, this is presented in An example of a ring Morita equivalent to the Weyl algebra $A_1$, S.P. Smith, J. of Alg, 73, 552 (1981).
Another result which may be of interest -regarding your fourth question- is that:
If the semigroup $k\Lambda$ has the same quotient field with $k[t]$, then $D(K)$ is Morita equivalent to $A_1$.
Here $K$ stands for certain subalgebras of $k[t]$ and $D(K)$ for the ring of differential operators on $K$. This is shown in: Some rings of differential operators which are Morita equivalent to $A_1$, Ian Musson, Proc. of the Am. Math. Soc., 98, 1, 1986
Finally, if you are interested in examples involving smash products with finite group algebras, i do not have some readily available but i think it is natural to look for such in the graded version of Morita equivalence.
I hope these are of some interest to the OP. Sorry in advance if these are too old and you are already aware of them.
P.S.: One more thing which might be of some interest with respect to the second question: The article Rings graded equivalent to the Weyl algebra, J. of Alg., vol. 321, 2, 2009, generalizes some results of Y. Berest, G. Wilson and Stafford, in the setting of graded module categories.