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.