First of all, your ideas about an eagle and wolf are pretty far-fetched

Unless you’re raising it from a puppy, at best what you’re realistically gonna find with a wolf is one (or more realistically a pack) that tolerates your presence and doesn’t actively view you as a threat or prey and sort of provides passive security by chasing things out of what it views as its own territory. The amount of time, effort, and resources (giving it food and such) to get much beyond that just isn’t going to be feasible in any kind of apocalypse scenario. If food gets scarce, you, your pets, and your livestock are absolutely going to be on its menu, and it’s probably not gonna be communicating with your dogs in any meaningful way beyond “I’m in heat, wanna fuck?”

Nor is an eagle going to be acting as a scout. You can probably watch its behavior for signs of threats that you didn’t notice, and maybe it will see you and your home as a source of safety and come back to you and refuse to leave if it sees a threat, but it’s not really gonna go out on patrol and actively warn you about what’s out there. They can be useful for hunting, but pretty much anyone who actually does any sort of falconry will tell you that every time they let their bird go, they’re really just hoping that it actually comes back and doesn’t just fuck off and they get to watch as all that time and effort they spent training it fly off into the sunset.

Dogs- absolutely, there’s good reason that we’ve been keeping dogs around since time immemorial.

Malinois is a solid breed, smart, strong, motivated, reasonably healthy and free of major health issues compared to a lot of breeds (like German shepherds,) smaller than some similar breeds so a bit easier to feed, and very trainable if you have the time and know-how. I have one myself, she’s a rare lazy malinois (which still makes her smarter and more energetic than most other dogs you’ve probably ever met) and she’s definitely got very strong herding, guardian, and protection instincts, they can also be serviceable tracking/detection dogs (police and such use them extensively for that,) my girl has even shown some rudimentary pointing instincts, and for not being a breed known for having a soft mouth (quite the opposite) she can be remarkably gentle and has carried some delicate objects around with her without breaking them, so she could do retrieving work as well.

So a pretty solid pick for a do-it-all breed if you can only pick one.

Ideally you probably would have multiple dogs to fill different roles, there’s a reason we have bred so many different types of dogs- livestock guardians, herding dogs, sight hounds, scent hounds, retrievers, etc. they all excel at different things.

And ideally you’d want enough to have stable breeding pools without too much inbreeding. This would be a lot to manage, especially if you’re also trying to train them to be useful, a large pack of untrained dogs is more of a liability than anything.

Your relationship with them is also going to be different than with a normal family pet, most of them aren’t going to be coming inside to sleep in bed with you at night, they’re going to be outside working, and in apocalypse scenario, probably turned loose to forage for themselves since you can’t just go to Walmart for a bag of kibble, and they’re probably going to be bonding more strongly with the other dogs than with you.

Cats are useful for pest control. Again, this isn’t gonna be your spoiled housecat who sleeps on your keyboard all day, this is gonna be more of an outdoor barn cat that probably wants little to nothing to do with you most of the time, but it’s gonna keep rodents and such in check.

Chickens, turkeys, etc. are a solid homesteading option, eggs, meat, feathers you could probably use for pillows, mattresses, insulation in clothing and quilts, possibly for arrow fletching if you have or can make a bow and arrows.

Goats and/or sheep for meat, milk, and wool, smaller and easier to care for than cows, and in many cases will do just fine foraging for food.

Probably worth throwing a couple donkeys in as well. They make decent livestock guardians and can be put to work as pack or draft animals if needed.

Rabbits are also a solid, fairly low-effort meat animal and of course have nice fur.

Beyond that, I can’t think of many other animals that would be particularly useful, desirable, or realistic to keep in a survival situation.