Like most things, neither side is completely 'right' here.
Neighbor shrugging off cat dumping complaints = Jackassery.
Shooting a cat with a pellet gun = Douchebaggery.
Like others have said, if you decide you really must shoot an animal with a potentially lethal device, make sure you kill it.
There was a certain neighbors yappy dog that loved to come into my yard and yap at me whenever I was outside. With two-acre lots, and a road in between, this was more than just an accidental crossing of property lines. My lab is unfortunately too mild mannered to eat it. After repeated failed attempts to get the owner to control it (she was like 90 and couldn't even catch it), I resorted to the paintball gun, which fixed that situation pretty quickly. I think anything that just leaves bruises (paintball / airsoft gun, slingshot, hand-thrown rocks, etc) is probably acceptable in certain situations.
Ha! Funny story time:
So my father, back in the day, likened himself a runner of sorts. Actually, her preferred the term "jogger," which to me sounds like a half step above speed walking.
Regardless, about once a week, he'd don his short-short-sweatpant cutoffs, headband, and sweat-stained T-shirt, lace his faded Niken cross trainers, and head out the door.
One day he came home, sweaty and breathing heavy as usual, but with a stone-faced ashen complexion. He gingerly sat down on the chair in the entryway, stared blankly at his shoes for a moment, and then spoke softly as he delicately unlaced them.
"Oh man...oh man..." he mumbled. Then, with a bit more authority. "Goddamn dog."
Father is a pretty devout Christian, so to hear him blatantly take the Lord's name in vain sent goosebumps shooting up my neck.
"You OK day?" I squeaked, as my tiny tot hands put a vice-like deathgrip on the teddy bear they clung to.
Dad shot out of his slouch and turned toward me as if snapped out of a trance.
"Oh...yeah, bud. No worries," he said, choking out a chuckle that seemed to catch halfway up his throat. "Just ran into that pack of Pomeranians again down the road."
Ah, the Pomeranian pack, or more commonly referred simply as "the ankle biters." Everyone in a country mile had experienced this unruly bunch. The quaint sky-blue house they resided hardly stood out as a nefarious hideout, but let me tell you: the Chicago penthouse suites of mob bosses held a more likable lot. I'd wake up in cold sweats, nightmares of the dozen tiny razor-fanged furlballs bounding after me, yipping and gnashing their jaws as I rode past on my Huffy, my small legs churning in slow motion, tires digging into the soft dirt road -- the pack getting closer, and closer, and closer...
"I kicked one," dad said, finally. "I didn't meant to, but I did."
"You what?" I said.
"Well, it ran up to me and nipped at my legs, so I squared up and punted it like a football."
My eyes turned wide and the corner of my lips curled. The image of one of those tiny bastards from hell spiraling through the air gave me more satisfaction than an entire weekend wasted watching cartoons and binging on junk food.
"No way! That's so cool!" I exclaimed.
"No son. I shouldn't have done that," dad said, in a very grownup manner. "I felt so bad, I actually went up to the door and apologized to Mrs. Anderson."
"How come?! Those dogs had it coming!"
"Actually, that's exactly what she said. Still, I'll never forget the look on that little dog's face. Oh well, hopefully they don't come after me any more."
Dad's brief pro football dog punting career didn't phase the ferocious furballs, but every time I rode my bike past that little blue house across the creek, I did so with a little more confidence and dangled my leg out menacingly if the Pomeranian pack ever dared get too close.