Some houses share a fence with cemetery - I get your point for the most part but where to draw the line?

The fence, obviously.

Cemeteries are often sad creepy dumps to start with, which cultures different than yours would find disrespectful of the dead. To me, OP would help integrate these places into the life of the community, honoring those who are interred there.

Actually, for an annual fee you can let your dog run off-leash in Congressional Cemetery in SE Washington, DC. I assume that the cemetery requires that you clean up after your dog. The one dog owner I knew who paid the fee certainly would have in any case.