Yes, they do. The thing that distinguishes lower animals such as dogs from humankind is not the presence of a soul, but the presence of a spirit. Humans are trichotomies, and in the case of a trichotomy, each faculty is reponsible for housing a certain, different kind of life, each "life" being distinct from the other in some way. The physical body is the house of our physical life, the soul is the house of our consciousness, and the human spirit is the intended house of the consciousness of God.
Now animals have no reason what-so-ever for having a habitable spirit and therefore lack that faculty, but they can and do display evidence of being quite conscious, and possess a relatively competent measure of will and choice as well. I would even go so far as to say that dogs in particular also manifest emotions...up to a point. So if a dog has a will or an elective desire for some sort of thing, and also takes the initiative to achieve that desire or choice, then those attributes are roughly similar to those found in humans, and that is one reason why a dog is unique among animals, in that it will go so far as to prefer to live with a human being than with it's own kind.
They have a simple, functional soul which houses their consciousness, but they are still far beneath the exalted construction of a man or a woman, which highly preferred being can potentially function as the very house of God, a privilege not even bestowed upon the highest of angels. Any arguments then against a dog, cat, horse, etc., having a soul, as that faculty would render them sacrilegious to the exalted status of a human being made "in God's image", are then voided by this distinction made between a simple dichotomy and a complex trichotomy, or us.
A further and ancient rub that surfaces against an animal having a soul is the question of resurrection. The scriptures do not clearly teach that animals will be renewed in their conscious existence in another, future world, and yet they do not declare that they will not be. Scriptural silence is never a sound argument for vacuity or absence of truth. God very much wishes for us to study human resurrection, and that is why it is taught in the Holy Bible and not animal resurrection, however the abscence of teaching on animal resurrection in the scriptures is accounted for under this former consideration, and that only makes sense from God's perspective.
"If dogs, cats, mice, were resurrected, then wouldn't that make them recipients of Christ atonement too?"
No, absolutely not. Pardon and resurrection are two different things. We must be pardoned and justified prior to resurrection, because we are moral agents capable of transgression and subsequent guilt, but animals are not moral agents and so do not need a spiritual Redeemer, as they have no sin pollution within their conscious souls, nor in any spirit faculty either as this is something which they don't even have.
