My kids never ask if they could keep a dog that I bring into the house, they have never gone out and found a pet either. When I bring a dog into the house they walk around as if there is no dog. I am attempting to do this with Bran the dog you can get from the dawnguard dlc. I play on the Xbox 360 so I can't use console commands.
I've never hired a dog in Skyrim, but I did have my children ask about keeping a pet.
I've also never married in Skyrim with any of my characters, so I know this can be done.
The pet event is random. It's one of the random things they do when you go home. They'll already be in the house with a pet, and when you come home, they'll rush up to you and ask about the pet.
The exact triggers are not known, and there doesn't seem to be information, but my play-throughs involve giving them maximum amount of gold every time they asked, giving them all sorts of random stuff from Dragonbone Daggers to dolls, and basically positive, nice interactions every time I come home.
It is possible scolding them and making them sad will cause them to not talk to you / trigger the pet dialog, since killing their pets will cause them to not talk to you also.
First is you have 2 kids you can only adopt 1 dog. If you already have a dog it wont work.
The dialogue to keep the dog is "random" so there are a few ways to trigger it.
Stay away from home with the dog for 3 hours in-game, Then go back and save it in front of the house. Try walking in, a couple of times, loading the previous save.
Dismiss the dog in your home (in front of your kid), teleport somewere far away, then teleport to pick up the dog again. Then offer the dog once more.
It is possible to adopt Sceolang too. I have him as a pet he doesnt bark as much and they never call him "stupid dog" so i would go with Sceolang instead of Bran.