I grew up in central Oregon and packs of coyotes were common. We also have packs around our home here in Wa. Here are some things I learned growing up there:
1. A pack of coyotes eats a LOT of food. When neighborhood cats and dogs start dissappearing, they have exhausted their normal food supply and are feeding on prey of opportunity. It is time to go hunting.
2. It is likley that some of those dogs that have "dissappeared" are not dead. If they were predominantly outside dogs and lacked a strong human bond(and allowed to run free), they may have joined the pack for companionship. Unfortunatly this influence that dosn't fear man may lead the pack to do things a native pack would not do such as close approaches to homes and pets and children. When I was 8 years old we lost our Collie to our neighbors 2 black labs that were allowed to run free and joined up with a pack of coyotes. That hit and run you described would be a good way to lure a dog away into the bush and gang up on it. It was good that your dogs were trained well enough to come when called.
I make it a habit of knowing my neighbors and their pets. If I see a dog running free and I can't call it to me to detain it or ID it as one of my neighbors, I will shoot it! If it is a neighbor's dog and it won't come to me, I go call the neighbor to retrieve it. If they are unsuccessfull or unwilling, I shoot it! Don't get me wrong, I love dogs and have owned a dog for as long as I can remember(As I type this there is a yellow lab named Sadie laying on my foot). I would not lightly deprive someone of their dog but A domestic dog gone wild is a danger.
3. Coyotes are typically pretty shy. The only reason you will usually see a coyote in the daylight is if it is hungry or has picked up some habits from domestic dogs. If you start seeing Coyotes in the daytime, it is time to go hunting.
4. If coyotes are routinely surprised, they will remain wary and eventually move elsewhere. You could identify and "mine" the trails and runways in the brush around your property. Nothing lethal that might hurt your children, dogs or your neighbors pets, just something unexpected. Tripwires to noisemakers, string snares tied to tin cans, fishing line tied between bushes, Caltrops(animal equivelent of a spike strip, multi pointed objects spread in fields to break up calvary charges). Routing gunfire around your property, particularly in the evening and when you hear the packs close by might also help to disuade them from approaching further.
Next time you set out by the fire, take the shotgun and the spotlight. As mentioned, they are quick, you will rarely get one chance let alone a second. My personal preference is a pistol and a flashlight. Unless the light is attached to the shotgun it is harder to illuminate and shoot singlehanded with the shotgun. My other preference for low light shooting is a red dot sight such as a Pro-Point or Aimpoint. You can use these with both eyes open without actually seeing the target thru the sight tube. You just need to be able to superimpose the dot over the target in any fashion and that is where the shot is going to go. A laser sight might also make target aquisition quicker in low light, but I don't have much experience with these to comment. As for your choice of ammo, unless you are close enough to hit one before the shot expands much(30'-40'), you are most likley only going to wound it with #4. You either want to hit it with a single larger calibre projectile or use larger shot.. I would reccomend 00 Buck(.22 calibre pellets).
A gated fence around the immediate yard might ease your wife's fears a bit particularly where the children are concerned.
Welcome to the jungle...
