You started this post, so I am going to respond, but I realy do not get it.
You are very allergic. regardless of stray cats, your neighbors will have pet cats. And, pet cats are outside some, alot or all the time and there is no effective difference for your problem whether it is a stray or pet, you are allergic to cats. Cats climb and jump very well, they go everywhere whether pets or strays. I dont get what the dillema is. If you are that allergic, you realy have no choice but to make your backyard as cat proof as you can.
[PERSONAL ATTACK DELETED BY MODERATOR]..the reason I cant hurt stray cats....
The reason one does not hurt stray cats is that it is wrong to hurt animals. We even have laws for food animals and hunting with the purpose to not hurt, to have a quick clean kill. We also have societal norms to not kill animals without clear need or reason. There is no correlation between the beaver story and the cats. The cats are not causing that kind of problem, you personally are allergic. The solution to stray cats is to catch and give shots and spay/neuter so that they do not carry disease ( see, the disease problem solved....) and do not reproduce and end up with starvation and suffering.
You havent addressed this normal, supposed best practices solution:
-- call and ask animal control about free or very low cost programs that may exist to get shots and spaying to feral cats
-- ask your neighbors how attached they are to the various strays, and likely they are attached and like them hanging around, feral cats are often not strays, but they are just not tame, often they are just feral, outside cats
-- broach the topic then of having the cats get shots and spayed, so there is no disease and no starving kittens later on
-- the cats then are back in the neighborhood, the ones the neighbors area attached to anyways, disease free and not reproducing.
Responsibility. Well, you are allergic. How is that any of your neighbors responsibility ? As they find out, they may learn to hold their dog back from licking you or not dump kittens in your lap, but I find that people have so much on their plate that they cannot seem to remember these things very well, the ones you are friends with at least should be able to. But, you can do things to minimize your chances of exposure. A fence. Cats kept from reproducing.