Do what the humane society says with regards to the trapping - they are giving you the right advice. You can trap overnight but you must be there to keep an eye on the trap at least every hour. If the cat won't go into this trap, you can build a drop trap relatively cheap. We cleared out a colony except for one big male that would not go in the trap - we caught him in 10 mins with the drop trap! And he is actually friends once you get him in...
The reason you put the food on the ground, under the trap, is so that when you lift the trap off the ground, the food stays on the ground. If the food/bowl is in the trap with the cat, the cat could get really messy or injured.
Canned tuna or cat food is fine. So are sardines. Get a water based, not an oil based. Basically you want something smelly.
Yes, you can still be charged for cruelty to an animal even if it doesn't belong to you, even if it's accidental (if there was negligence, of course). Usually this is in cases of willful cruelty.
If no one is owning up to owning the cat, he may be a stray. Can you tell his condition - is he skinny or dirty? If so, trap him and bring him to the humane society without a worry. It will be best for him.
Why did you remove the cardboard cat craft? It was not attracting the cat... He considers your yard part of his territory. I also think he may indeed not be hunting your rabbits because he wasn't gone after them yet. Cats hunt around the clock - if he wanted to, he could have by now. Not that I am saying to take the chance or anything. God I wish people would keep their cats indoors. However ---- there is something called the vacuum effect. When you remove a cat (typically a male, or a colony) from an area and rehome them (or euthanize them), you will find that new cats come into that same territory a relatively short time later because the territory has opened up.
I know you are sick of hearing this but I have to say that I think you are severely underestimating the danger you have put your rabbits in by leaving them in an uncovered back yard, unsupervised, for long amounts of time. Just google "coyotes in the city" and look at the pictures - coyotes trotting through the suburbs during the day. On busy streets at night. INSIDE stores and in the subway (RIDING the subway, on a seat, like a person). Look up "foxes in the city" and you will see the same thing. Raccoons we all know thrive alongside people - they are small and cute but fierce predators, and smart, able to open doors and reach through chicken wire to grab animals, and brave enough to kill relatively large prey. Overhead predators may also exist in your area and while probably unable to fly off with a large rabbit, could injure it or fly off with a smaller one. Plus stray dogs and cats which can jump into or dig into your yard and get at your rabbits.
I would hate to know that you lost a rabbit to something so preventable. You feel safe in the city - most people do - but you're still in the wild and there are still wild animals there. You just don't see them very much.
I know you say you want to let them out in the yard because you don't want them to be stuck a run, even if it's big. Your yard isn't the wild. It is nothing more than a large run, except it's unprotected. There are many ways you can keep your rabbits safe in a run while still allowing them ample space, and time outside when you are around to supervise.
You haven't even lived there a year yet. People who lose pets to predators do not always do so within the first year. It can happen years, even decades, after they have settled in. And no one expects it. And if a predator gets in your yard, especially a fox or raccoon, it will kill all of your rabbits if it has enough time to.
Of course, they are your rabbits and it is entirely up to you, entirely your choice. But don't fool yourself into thinking that your rabbits are safe just because you live at an intersection and you've only seen one predator in the eight months you've lived there. Your rabbits ARE at risk - it's fairly small but it's not tiny by any means (as in, I am not expecting it to happen tomorrow but I would not be shocked if it did).
PLEASE consider building an enclosed run for them for when you're not home... They will be safe and you know that they are even when you aren't around to see them.