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)][1]. 

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][2]

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. 

  [1]: https://pdf.sciencedirectassets.com/272332/1-s2.0-S0021869300X04140/1-s2.0-0021869381903343/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEKn%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIQCU7nuDMGJ03UKrn5bQrthYdY9hqLxlNx5tfxfO7EbRBQIgFagWGmwJ71Qv7bAnEsDHHALo6qpRUn4FPLCvY2ne9jcqtAMIYhADGgwwNTkwMDM1NDY4NjUiDFKgeWvyvPssCMQ6SiqRAx%2B2ySUd25MQ13h1NOZJz%2BiOoZMGkRDGyAwW%2BIhDAg%2Bem5WfW%2BAVK434RJCrRSlzp6SvFfIx%2BSTuD%2FHe3AEhfZkPdh8YbvNF%2B56FFwa1XljMofFME%2Bbc7rHi1iSzg84O0R%2BufKU4obH7SLyb0hJjO817UZc1PkuEO%2Bpk7lpLfO5MeTAtJuomSWnJ%2B0SNoMSEd4Z8h2lpK%2BpBaNMB8gesJI5MXpbAe2zuexRzx1lKvkMJVY%2BBJukf2qFWLWOlA2GhAmilN86yMVVqQ2DHNkDv%2BARtjXu%2BAiz2EUXAsuNLn6azAEsUxaiy907Zic9Ez6ghKrOvzawgXvyhlnacS2B2r7VOo9YgM7onx6aoJyPO7iQV8E2jgCLlUXwPm5vjmzrfzOWyJAat8EmdLIBXLUvyACjmGYCPve5kXU4Wih82m6gYCRXMUwlmuty16QumyovDsy3uvrawQXRSfwEbjprvjYH%2B%2FhkDHtQmtk%2BdA%2Fdz6IefFrkVKwjkx%2BDZvtVRk8vC3k6PKevucS3EkAWKINkfmhnLMMzU4fgFOusBbhF7qZyMmrNn23uaL3ffpBztNFUag8ksrp8qbiaJ6qFyVjigmFn2BWYjugzBSoyA5qixy6pJ0gj41ZgSZVXw0h3fQL9vftDiQbXANAZtlEF3eX8JREyL7qEx00X4WXPyNTxkkJh0CFMC8hDEHSDxVvX2zQmZgCEpGsMe3VsSiKPK6FofhjgkWfuF2PfUrTBO6WL6cGxOucuvYtdXJOLWvjUcDEmrkdbTJiqPILumVawfs3KAAzo7KJsb2AN8n1WM2oSVI7oH3hsAbEl%2Fx7ljANBC%2F%2FJ%2Fmbifho5i8lqGG3GSF%2BT%2B26gKIwXymw%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20200722T175345Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYWMRXLSUN%2F20200722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=e42d4d7efd46168a90cb275debe065cc559bfa0f82d9988c83c0538944adcd41&hash=219169c646b6c74c71403124b515e9b4b61ac091d0fff0c241183bb2746fe852&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0021869381903343&tid=spdf-6803cfbf-d67b-49bf-b72b-5e96ece41f05&sid=76b8711923cd114cdc8aa9c4d50e1ff58851gxrqb&type=client
  [2]: https://www.ams.org/journals/proc/1986-098-01/S0002-9939-1986-0848868-1/S0002-9939-1986-0848868-1.pdf