In Genesis, it speaks of creatures having souls. So, yes, I agree with you that animals have souls.
Our soul is the living part of our being; the part that goes on after the physical body has ended its life. Our soul is the most important part of our being that we should take care of. If our Soul is healthy, we are healthy, even if we have physical affliction.
Lover of my Soul!! That is the person I want in my life!
Love Always and God Bless!