As to souls, well; I have known animals who'se personality and friendship I treasured more than most humans I have met. While not a dog person (I am a cat person), I recognize those traits I love in my cats are also in dogs (however, I must proclaim cat superiority, they would if they could type). I would hope that when I pass God would grant that I would be reunited with my special cat friends who have passed before me.
The answer is yes. However, they don't have 'immortal' souls. The definition of the word 'soul' according to the theological aspect is 'that which gives life'. The soul is the animating and vital factor which leaves at the time of death.
Humans have immortal souls.
STAY AWAY FROM MY DOG.
Well, I don't have a dog yet. But I will. AND THEN YOU STAY AWAY FROM IT.