First of all I am SO sorry about your mom's cat. I volunteer with a local feline rescue. We try so hard to explain to people that even if kitty wants out you have to keep them in. We have several reasons why, among them are: wild animals, poisons, diseases, dogs, neighbors,... it goes on and on. I think the dog owners were in the wrong but if kitty had been indoors this wouldn't have happened. We do not adopt to people who let them outside.
Yes your dog can run free inside an invisible fence if it has been trained to stay in your yard and you keep the collar on and your yard is marked. The problem is the stray dogs that come into our yard. Our dog is only allowed outside the fence with a leash on and yes I pick up her poop when I walk her. Pet owners have to take resposibility for their pets.