I would think a gym would be an occupied structure. That is a change in use. It is possible that a garage or storage shed might be permitted within a setback, though an occupied use would not be permissible.
I am not an attorney, nor am I familiar with your local area, so a local land use attorney would be better suited to speak to the probability of getting the structure torn down.
While I doubt that you personally can limit future construction, if an existing non-conforming structure is torn down, it is unlikely they could rebuild. And if they did try, they should need to go through zoning. As a neighboring property, you should receive notice of the zoning hearing board meeting at which such a variance would be considered.