I think the company named Topaz had a photoshop plugin to remove "jpeg artifacts" - I don't know if they are using a neural model for it though.

I did it some time ago. It works at low levels. Doing it in hard cases is non-trivial but possible. It's difficult to monetize that's why nobody is doing it seriously.

Yes, there are plenty of them. Not sure what the SOTA is, though. Similar to super-resolution, it is relatively simple to create a nearly-infinite dataset for these; pick a clean image, then introduce JPG artifacts. Then train a model to invert the process.