You didn't mention raccoons in there. I can't believe you have all those other critters and don't have raccoons. They are the worst pest in my garden, totally destructive.
As noted the only problem with cats is if they use your garden as litter box. We have three outdoor cats we feed, but have not really had a problem with that. Also as noted squirrels will do anything for corn, but otherwise don't seem to bother the garden much. There's tons of squirrels here and I don't have trouble with them, except they do love the bird seed. But we have the bird seed in squirrel proof feeders now, so the squirrels just help clean up what is dropped on the ground. Our possum likes the dry cat food we put out for the cats (as do the raccoons and bigger birds like starlings and blue jays), but otherwise doesn't seem to mess with the garden too much. The birds are beneficial in your garden.
[I also gave up on growing corn, even when I used to live where I had space enough to grow it, because EVERY outdoor critter loves it, birds, squirrels, woodchucks, raccoons, mice, rats, voles, shrews, etc etc....]
So that leaves the woodchuck and the raccoons if any, both of which are destructive. I have discovered that the woodchuck thinks sunflower plants are a special treat. Any time I plant them, he comes along and bites the tips off the plants when they are only a few inches high.
Anyway really only two things to do, barriers or repellants. Without building a fence, deer netting (also sold as bird netting) works pretty well. I use it for my tomatoes, which otherwise would be totally devoured by both the woodchucks and raccoons. It's very light weight and easy to throw over things (but you need a frame, if it's just resting on your plants, critters can bite through it). But the netting has to be VERY well secured at the bottom AND the top. I had it tied together at the top with plastic twist tie kind of stuff and the raccoons UNTIED it!
Repellants that people recommend (I haven't tried it, I've been going the barrier route, but probably a combination would be best) include human urine, human hair, bars of Irish Spring soap hung around, epsom salts, ammonia soaked rags. They sell things like coyote urine as critter repellants (I always wondered how they collect that!
) Also cayenne pepper on your plants is repellant for critters and insects.
I don't have rabbits in my garden, and I'm not sure if the rabbits could bite through the deer netting. If so you might have to add 18" or so of chicken wire...
You can also, as my friends have, pretty easily put together a home made solar powered electric fence. I'm thinking about that for next year. You only need two or three strands of wire, maybe only two if you don't have deer in your garden--you didn't mention them. They are the most destructive garden pests of all and getting more common in many city as well as suburban areas. I haven't seen them in my garden (yet), but they are around just a few blocks away.