Your coop and fenced in area isn't safe either if a critter is living in there, it is obviously coming and going in and out of there somehow. Bottom line is, if you want to keep chickens you have to have either a completely secure area to keep them in or have another plan in place that involves trained dogs, traps, guns etc. even then you will lose a few here and there. Keeping an eye on them while they free range does not work, predators can and will come in and snatch them without you even noticing. They are very good at what they do!
I would keep your chickens confined, even if it is cramped, until you get the neighbor's dog situation under control one way or another. But as I said before, even then you will be facing a multitude of other predators because chickens are easy prey. Best of luck to you.