Cartoon History of the Universe discusses the concept of selective kindness in talking about inter-cultural relationships in the days before diplomacy: everybody tender to their own people while at the same time callously brutal to everyone else.
Dorian Chief, doting over his wife while being surrounded by piles of corpses: Ooh, honey, did you hurt your finger?
Kingdok from Bone prepares and brings food to the two stupid rat creatures for inadvertently doing him a favor. Unlike most examples, Kingdok actually retains most of his scariness and menace as he's doing this.
Lobo of The DCU parodies this: he is a relentless killing machine who often kills his bounties instead of capturing them, yet at the same time has a soft spot for "space dolphins". But he did once spare Aquaman's life because he felt he couldn't hurt somebody who loved dolphins as much as he did.
Vril Dox has a couple of these in L.E.G.I.O.N. Right before he is raped and killed, he confesses to his unconscious teammates that he really does care about them, but he doesn't really know how to express that. When Lyrissa Mallor dies, he mourns for her, believing that without her, L.E.G.I.O.N. would not have gotten as far as it did. When Vril Dox finally gets to see his son, after Stealth believed that he would become just as controlling as he was of the rest of the group, Vril remembers his father being a controlling jerk, and himself being controlling to the Durlan. So he lets the mother have the child. awww.
Catwoman obviously can't pet a dog without ruining the motif, but she's been subjected to a number of Pet The Cat moments, a prime example being the ending of this web-toon. She's not the least bit disappointed that she wound up saving kitties instead of stealing gemstones.
Edward Blake in Watchmen has a couple of moments like this. The first is when he is talking to a teenaged Laurie and tilts her chin up to look at her eyes, which he says are like her mother's. It turns out that she's his daughter, and he looks a little sad when Sally Jupiter drags her daughter away from him. The second instance of this for him is when he cries after finding out about Adrian Veidt's plan to achieve world peace by killing millions worldwide. He even cries when talking about this to an old archenemy, who he says is "the closest thing I have to a friend." Considering that Blake has several instances where he could be said to cross the Moral Event Horizon, these moments are probably sorely needed for his character.
Hellboy is a huge demon with enough strength to tear a car in half. He is the Badass to end all badasses. As such, his action figure comes with the following props: A six-pack of beer, a revolver, and a... kitty?! Yes, the son of Satan is a cat-lover.
Also in the film Hellboy II: The Golden Army, Prince Nuada had a literal pet the dog moment. Justified by the fact that his big problem is with humans overrunning the planet and pushing out other species, so it's natural that he's a lot nicer to dogs than humans.
Artist Katie Cook  draws figures like Darth Vader and General Grievous playing with kittens in her Star Wars: Clone Wars comics.
In Negation, the Saurian party member subverts her pet the Kaliman retriever moment when she names him "tasty treat" in her people's language.
In Punisher War Journal, the Rhino has a few. First, he writes an apology letter to the widow of a cop he had accidentally killed and sends her money with each score he pulls off. Second, in the last issue he convinces the Punisher to leave the Stilt-Man gang alone (on Christmas no less) because he could tell they were just stupid, not evil.
Dr. Octopus from Spider-Man has gotten a few pet the dog moments over the decades, including trying to save Aunt May from Hammerhead and trying to develop a cure for AIDS in order to save his ex girl friend.
In the climax of the Drowning Man arc, Deadpool shows us that he's given up on redemption by locking Blind Al in The Box, severely beating Weasel, and... petting Deuce the Devil Dog on the head.
When Gotham City is destroyed by an earthquake, usual Batman villain Poison Ivy takes over the city park and makes it a haven for the city's orphaned children, caring for them like her own children. When the police see her as a dangerous terrorist and try to take back control, the herbicidal chemicals they use poisons one of the girls inside, and Poison Ivy chooses to give herself up to save her life.
While Emma Frost of the X-Men is still a bitch after joining the team, she gets more than enough Pet the Dog moments around X-23/Laura Kinney. While at first seemingly hostile to the younger girl due to her being a female clone of Wolverine that was created solely as an assassin. It's eventually revealed that this is because Emma see's the same capacity to hurt others in Laura that she had when she was younger with the main difference being that the latter has no choice in the matter due to her conditioning to a "trigger scent" (which sends X into a berseker rage). Later on, we see Emma eventually warm up to Laura to the point where whenever whenever we see Emma in one of X-23's books, it more often than not involves her lamenting on how they've failed to treat her right.
In the beginning of The Punisher MAX comic Kitchen Irish, Frank is eating lunch in a diner when a bomb goes off in the pub across the street. People all around him get butchered when the diner's large window shatters, and once the debris settle he finds a man whose chest had been shattered, revealing his heart, giving the following Pet The Dog moment from Frank.
Injured Man: help me
Castle: ''So I do. For no reason I can pin down it becomes very important that this guy makes it. Maybe he has a wife and kids. Maybe he wants to see them again like nothing else on Earth.
Frank works on keeping the man's exposed heart from bleeding out, and when a rookie EMT gets there they spend an hour working together on the man before finally succeeding in stabilizing him. It takes several more minutes for the traumatized EMT to notice that the man who helped him save an otherwise-doomed life did so with six inches of glass sticking out of his arm - and that he is a mass-murdering vigilante. He treats the wound without another word, and an understanding is silently reached; the EMT will go on saving lives, and the Punisher will track down whoever started a war in downtown New York.
Castle: (Not every day you meet your polar opposite.)
Negan the big bad from The Walking Dead is a sadistic monster that likes violence but in issue #105 instead of killing Carl for shooting at his men, Negan instead had a civil conversion with him and he even apologized for hurting Carl's feelings when he brought up the issue regarding Carl's eye.
Orm in the New 52 Aquaman book is noticeably less evil than before Flashpoint, but he regards land-dwellers with disdain. When the prison he's kept in is destroyed, he leaves, but not before murdering one of the guards who was nice to him as thanks (which he does seem to regard as a mercy considering the man was wounded and likely was going to die a more painful death at the hands of the prisoners). He meets a woman, Erin, on the way to the sea, whose son is being attacked by escaped prisoners. He disregards her and tells her that when he was Tommy's age, 8, he had learned to defend himself, and it's her fault for not teaching him. He leaves and goes back under the sea, before returning to help them while admitting that 8 is too young an age to die. Later, it is revealed that he stayed with them through the Crime Syndicate's invasion of Earth.
In "Spider-Man Family#4" the Puppet Master of the Fantastic Four meets a young boy named Louis, and not only comforts him when he's picked on by bullies, he gives him the Spider-Man doll he was making to use in one of his evil schemes, willingly abandoning one of his evil plans to comfort a distraught boy he has only know for a few minutes.
In MAD, this is parodied and even deconstructed in an article on "Compassionate Conservatives." The "compassionate" parts of the conservatives are typically described as hypocritical (deporting all illegal immigrants except for their gardeners) or inconsequential (giving last meals and a choice of an execution method to death row inmates while denying them appeals on DNA evidence).
Spider-Verse does this for the Spider-Man Newspaper Strips universe by having the Master Weaver use Comic-Book Time to pull Morlun away from that universe and claim that it was temporally unstable, all the while shunting the universe into a pocket universe, sparing a universe where Peter and Mary Jane are still married.