That's the main reason I had security screens and doors fitted at my place - not so much to keep out intruders but to keep my two kitties in. They are indoor cats only, have never been outside unless crated. Too many things can go wrong if allowed outside ie, straying, fighting, encouraging other cats into the yard, hit by a car, stolen, hurt by irresponsible neighbours or their pets (& I have one of these unfortunately), catching fleas and most of all disease. I do feel so sorry for the poor kitty, but most of all I feel for you. I know that neither you nor your dog are responsible for any of this and I hope your neighbour realises this too. I also hope that your neighbour realises that you already feel terrible about the loss of the kitty and hope that they don't add to your distress. Hope the health of you and your doggie improves quickly and well and nothing adverse results from the cat bites. Try and keep calm and take care.
Go and have a cuppa!!
Cheers