Sorry to hear about your loss and the trauma for you and your family. How wonderful that 3 escaped.
Legally around here, as awful as it may seem, the land owner is fully responsible for fencing other animals off his/her own property. Your idea about establishing a fenced area where stray dogs cannot get to your chickens is really the best solution. It may not fly in court, unfortunately, to try to get some reimbursment even if you had solid proof of exactly which dog was the culprit. A lot depends on local ordinances, courts, lawyers etc.
Regarding going to neighbors, I do agree than no one would claim responsibility, and neighbors may not even know where their dogs travel.
All that said, it sounds like there is a problem dog in you neighborhood. Not sure where you live, but perhaps you could call your local 'animal control' office to have the dog that chased your children and killed a cat picked up and removed. Tell them the facts and suggest that it is a public nuisance, perhaps even a public health/safety threat.
This thread didn't mention the idea of feral dogs either. Occasionally there are dogs around that really don't belong to anyone, they have been abandoned.
Your solid fencing around your A-frame will probably be the best and cheapest defense in the end.