Municipal Animal Shelters euthanize the majority of cats that they take in. Methods vary, depending on location / state, but are generally pretty horrible. Gassing, sucking the air out of vacuum chambers, lethal injection, and shooting are all practiced. Veterinarians are generally not used for this - too costly. If the poor cat is to be euthanized, it would be a kindness to at least take him/her to your vet, have a sedative shot administered, and then the lethal IV solution. This is a much less awful end then a shelter can afford to do.
As for fighting with your cat, if you did want to introduce them and have them get along, it most likely can be done. A good way to do it involves bringing the cat inside, and putting him into a separate room for a period of time. This allows him to get used to the environment and smells. It allows your cat to get used to him too. Introduction must then be very gradual, First smell and hearing, then limited vision through a small opening, etc etc.
There is an art to this. There is a lot of info available on how to do it. Take your time, and it most always works out well. Many people make the mistake of rushing it, especially when it seems to be going well.
I have been involved in cat rescue work for many years...
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