Oh good, story time. So... That's not entirely true. A neighbor of mine used to have a grumpy old male cat named Kiki. This cat had seen some serious shit. Had the typical anime-scar on his left eye, and a glare that could cow a pitbull. We assume his name was the source of his eternal ire.

One day, a hawk starts living nearby, and the neighbor's cats start going missing. They lock their doors and keep their cats inside, but cats will be cats, and one by one they get out and are slain by the hawk.

Finally, the day comes when Kiki is nowhere to be found. Everyone fears the worst, and they run out back to find a contented Kiki not far from a pile of feathers and a beak. Never found the rest of the skull, or any other bones, mind. Just the beak. Hawk never stood a chance.

There have been nest cams that caught eagles feeding their babies cats before. It definitely happens. I cannot attest to how common/frequently it happens, but it does happen.

They don't have to carry it away whole to kill it. Our eagles seem to all kill fish (live on a river), but I wouldn't put it past them to gut a cat or small dog if unattended, and if the eagle was hungry enough.

Depends what eagle it is. They didn’t mention species or country, but there are certainly eagles that would both live in your yard and easily carry a dog.