Sorry for the confusion.
No, you are NOT doing anything illegal as far as I can tell. People who trap and relocate,
without the proper permission and such... they are. But worse than illegal... they think they are doing the kindest and most humane thing for the relocated animal... not realizing they just put it in a fight for it's life. So I try and put the information out in case someone reads it and makes the realization.
Yes, by dispatch... I mean to eliminate by humane method (such as shooting). Not the same as "harm". It's what most of us in the forest must do in order to stop them from coming for a free chicken dinner night after night. Again, I personally use containment of my own animals when needed. Hawks are my biggest threat since my dogs keep daytime ground predators away and my coop is locked up with any holes covered in 1/2 inch welded wire hardware cloth for night time lurkers. Raccoon, domestic dog, opossum, weasel, and many others can do a LOT of damage to a flock.
No, it sounds like the shelter was honest with you about having to use euthanasia when necessary! I just put it out there for people who may not understand that. I don't think you have done anything wrong!
While there are laws in place for people to keep their domestic cats on their property... I have YET to meet anyone who has a cat with outdoor access and ensures they stay on site. I would NOT feel bad about trapping a neighbors cat and taking to the shelter. If they have to pay a bail... maybe they will be more responsible owners... if they are smart enough to check the shelter. And those cats are PURRfectly happy to claw up the paint job on my brand new truck or car to get on top of my warm engine.
How old are your chicks?