Okay, so, to preface this- I do know dogs have already been marked a 'yes' in the "common suggestions" thread, so this isn't a general "please add dogs" suggestion thread.
Instead, I'd like to introduce some particular ideas I had for the implementation of dogs into the game, and how they could function in a way that makes them interesting.
1. "Follow Me" - Party System
So, one of the big things here is- ideally, if you have a dog in a game like this, you want them to be able to travel with you as your companion and make the game just a little less lonely. The idea here is, when you have a dog set to travel with you, it gets its own stats panel that you can keep track of on the HUD, kind of similarly to your party members in an MMO or RPG. That way you can keep track of its health, stats, hunger, and so on, so you can effectively keep them alive. If your dog is trying to help you fight off a zombie horde and sustaining a lot of damage? You could command it to flee and save itself. You as the player have the capability to run away- so your dog should, too.
2. Dog Stats and Training
Ideally, to avoid dogs either being entirely invulnerable or super squishy and easily killed, they should have some kind of trainable stats system similar to the player. For example- strength, fitness, agility, etc, anything else that would fit and be useful for a dog. There could also be a "loyalty", "bond", or "obedience" stat, which affects how likely the dog is to obey your commands.
Dogs could be trained to raise stats, or trained in specific commands, that allow you to sort of control how your dog behaves while travelling with you. For example: Come, Stay, Attack, Run Away, Hide Under Car, Get In Car, etc. Maybe you could even train dogs as livestock guardians if more predatory animals are added or if zombies target livestock.
3. Idle Behavior
When in Travel Mode, dogs could have a certain radius around the player where they may still wander a bit. i.e. walking around, sniffing at things, and generally just... behaving like dogs. In general, they'll follow the player, although it may not always be right at their side.
When not in Travel Mode, dogs could wander freely(unless shut inside a house or fenced in), but have a sort of "home territory" they'll always try to return to and mostly stay within. The player could set the dog's "home", and then the dog could expand its territory radius from there via exploring, until it hits a certain cap and it'll mostly stay within that area. To keep dogs confined, you could keep them indoors or behind a fence. There could be a button to call the dog from anywhere, and if it's within a certain radius, it can hear your call and will come to you if trained- but of course, this call could also attract zombies within the same radius.
4. Multiple Dogs
Would be nice if you could have multiple dogs, maybe up to a certain cap, but can only bring one or two with you in "Travel Mode" to prevent it from being too OP. This could allow you to breed them, or train different dogs for different purposes. Dogs could interact with eachother when idling, just to add some flavor.
5. Dog Armor
The ability to loot things like K9 police vests and such to put on your dogs for protection, similarly to clothes on the player character.