I've not seen any evidence of animals having souls.
Gen_2:7 And the LORD God formed
man of the dust of the ground, and breathed into his nostrils the
breath of life; and
man became a living soul.
Gen_6:17 And, behold, I, even I, do bring a flood of waters upon the earth, to destroy
all flesh, wherein
is the breath of life, from under heaven; and every thing that is in the earth shall die.
Gen_7:15 And
they went in unto Noah into the ark, two and two of all flesh, wherein is the breath of life.
Gen 7:21 And
all flesh died that moved upon the earth, both of fowl, and of cattle, and of beast, and of every creeping thing that creepeth upon the earth, and every man:
Gen 7:22
All in whose nostrils was the breath of life, of all that was in the dry land, died.
Gen 7:23 And every living substance was destroyed which was upon the face of the ground, both man, and cattle, and the creeping things, and the fowl of the heaven; and they were destroyed from the earth: and Noah only remained alive, and they that were with him in the ark.
Every living animal has a soul. Without the soul, the earthly body has no life.
If children who are innocent go to heaven when they die without being born again, would it be different for an innocent animal?
1Co_7:14 For the
unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else
were your children unclean; but now are they holy.
Why are the children holy if one of the parents is a believer? Though they are not too young to believe, they are too young to understand the gospel.
Why is Abraham called our father? We are his children because we have the same spirit of faith in us that he had in him. And how did we get this spirit?
There is something we call transference of spirit. Some of the ways this happen is via one looking up to another, being under the covering of another, or agreeing with another. And dogs certainly fit all three.
If an animal can receive an evil spirit in it, it can certainly receive a spirit of or from God.
I don't know, but I think it is possible for our pets to go to heaven.