We have 2 female indoor-only cats, both spayed. They don't go into heat. But we had a problem with male cats coming up on our front porch and spraying the walls and door, and sometimes the back door on the deck. The smell seeped into the house, clung to our shoes and got tracked into the house and car. One morning when I got to work I had to straight to the restroom and wash the bottoms of my shoes because I reeked. They ruined the rugs on my porch and a large wicker basket. We were constantly washing the porch and siding off with bleach water. DH was fed up and got traps from the pound. He caught 3 and took them to the pound. One belonged to next door neighbor -- DH said he didn't recognize it, but I think he was just so fed up he didn't care. That cat was neutered and I actually don't think he was the problem. They apologized for the inconvenience to us and I offered to help pay for the $200 fee they had to pay to get the cat back. They declined, and things felt uncomfortable between us for a while but it all settled down.
Another cat was a huge ratty male with a torn-up ear and in very bad shape. The pound put him to sleep because he was so unhealthy. A few days later a lady we didn't know stormed into my husband's office and threw a fit about him being a cat hater and killer. She lived down the alley behind us and had been feeding that cat, along with several other feral cats, which were likely the ones we were having problems with. She said she'd heard we were trapping innocent cats and hauling them into animal control and she was livid when she found out the cat had been put down. She was totally unreasonable. DH told her in fact we love cats, but we are responsible for our animals. If she's feeding them and encouraging them to hang around, she's part of the problem. And he said if cats continued to spray our home, he would continue to trap them. Animal control would figure out what to do with them. Once that ratty cat was gone, we had only a few isolated incidents of spraying.
If the problem continues in your yard, continue to trap the cats. You might want to try the mothballs or other repellants as well to keep them out. If you want to take the high road, call your neighbor to identify the cat and give him an opportunity to reclaim it before you take it away, but he sound totally unreasonable. Document any harassment as well as every time you catch a cat and what you did with it. You are within your rights to take care of your own property.