Annoying old neighbor is offensive and rude and trespes... what to do?

That sounds terrible. I think you might have to confront her, in a nice way at first. Say that you really enjoy living on this block but you are feeling overwhelmed by having to talk so much to all the neighbors, and please don't come on to your property any longer without being invited. You are just too busy with school or whatever, and need to concentrate on that more. Just make up some reason. If she keeps doing it remind her that you asked her not to come on your property. If you have to walk over and she tries to talk cut her off right away, say you are in a hurry and will talk another time, then keep on walking. Fast! Good luck!!