I'd open and close the door as normal; if a stray is in there, so be it. If you don't want extra birds, I would call/send a letter to let them know and, if it continues, off to the freezer it would go. As to the fence, I also have neighbors who don't fix their fence (or anything, for that matter). When we were looking at the hole with intent to repair, the lady came and told us it was not ours. Why would she care if I fix it? So, I have decided to leave the hole. If a dog (another neighbor's, just young and not trained) comes in my yard then I'll take it up with him or set up a temporary fence. I"m thinking of the kind you drive the posts in the ground and run wire, could put it up on my side of the holey fence and keep the dog out.