This is the same thing that the Mormons teaches. I don't know anywhere in scripture that says we will become God. In fact, going back to the first sin the deception of the serpent was simply that we would become "as God". And Satan's fall was over the fact that he wanted to be like God. The tower of Babel was destroyed and the languages confused because of people wishing to be on par with God. I think there is evidence that this is the greatest sin; to think that we will be like God.
The Bible does say we will be "sons of God" in several places (e.g. Rom 8:19, Gal 3:36, etc) but I think it is a mistake to equate that to being like God the Father. I don't believe that is even equivalent to being the Son of God, as is our Lord Jesus, who has the name that is above all names and remains our High Priest. The scriptures clearly designated us as "sons of God", and there certainly are privileges and responsibilities with that title; but I believe its a grave mistake to assume we will be like God the Father. I'd certainly feel more comfortable using the "sons (small "s") of God" title that He has given us.
I am surprised you mentioned animals not having souls. Do you think the soul is immortal, HD?
Yes, the soul is immortal. Do I believe animals have souls? Off hand I would say I don't. I don't believe in the spirit of animals roaming around in heaven. Fuffy the kitten most likely will not be in heaven contrary to what a vast number of animal lovers would like to believe. But this is an area that I'm just not sure as I haven't studied this. I'm opened to being convince otherwise. It's not that I'm right or wrong; its simply that I don't know.
I did try to find a discussion of this topic but there seems to be scant writings on it, if any at all. So far I have come up empty handed.
“I don’t know anywhere in scripture that says we will become God.”
Its (theosis) a hard concept to explain in English. Try reading +Athanasius the Great On the Incarnation. Its full of scriptural references. In the meantime, look at John 10:34-35, I Cor. 15:28; I Cor.12.27; Genesis 1:27; Rom. 8:15-17; Rom. 8:29; I Peter 4:6; John 14:12. The list goes on, HD. For me, I think Genesis 1:27 lays it right out and fully explains what is meant by the later scriptures. For us to become by grace what Christ is by nature is the whole point, HD.
“Yes, the soul is immortal.”
Oh, I was afraid you would say that. That, HD is one of the earliest and most deadly of heresies. All sorts of error stems from it, as Origen, among others found out. The human soul is not at all immortal by nature. Immortality is a gift offered by God which we can accept or reject as rational and free creatures. I thought that Calvinism might, given its theology, of necessity hold with the idea that the soul is immortal.