I am sorry to hear about your dog, I don't know what to tell you as far as strategy goes without actually being there myself to give it a look see, but if it is possible maybe try to sit up in your house, those dogs are wiley and notice every little detail that is out of place, but they "know" your house is there...
I would recommend using a prerecorded tape right now, and as far as sound I would go with a dieing rabbit, since you have reason to believe the coyotes are getting all of em...
the more you listen to the tape and see how the critters respond to it, the easier it will be to take up mouth/hand calls in the future. Another sound I would consider is some sort of ground bird, quail,etc...
I would not rule out a bob cat being part of your problem either, and they will respond to the same sounds as a coyote, only they are not always as willing...
Also, unless there is an overabundance of fox, I would think twice about taking them, they keep the rodent population down dramatically, and I don't think they are the ones responsible for your dog or the rabbits, but they will respond to the same sounds as well. Just my opinion.
Stay Safe. Happy Shooting/Hunting.