Good deal on your hunting “pets”! I saw the picture of your Aussie in your machine shed.
My daughters Cat is a house cat, neutered and totally useless for vermin protection. when I can get him out he watches chipmunks and squirrels without moving He does occasionally chase birds. Inside he sniffs bugs. Useless.
I have chicken wire on the inside of the picket fence in the garden section of my back yard. I think they sneak in on the street side fence side under the fence. The ones I find in the garden are the small ones rather than big ones. I put up some more around the perimeter of this particular raised bed. Except for the peppers I do not think they will bother anything else (tomatoes okra, squash.) When the compost heap was outside the fence I would just about always find a rabbit litter when I was turning it.
Thinking about a garden and you are not on an urban rooftop? Part of the planning involves fencing or a wall around your garden. And if you are prepping and plan to move to the country and intend to garden, make certain to stock up on things like hog panels or 6 foot rolls of fencing and 3 or 4 foot chicken wire fencing or you will not have a garden for long!
For anyone needing info on Fencing See Pollards prepping links:
Info on garden gate and links to installing fences
And this:
Remember the 1" chicken wire around the base of your fence to keep the rabbits out! Bury it down in the soil, rabbits dig!