I am sure they have existential questions... I had fostered a dog once. Got him treated for worms, wounds and had to put a cone around him. He would spend hours staring at a wall. I'm sure he was miserably uncomfortable.. Poor thing... But made me wonder don't they question life at that point?
We think we have a sense of reasoning, awareness and animals don't so they can't reason about God? Do we ever wonder the animals are much closer to nature than we are. Maybe we have got it all wrong? Maybe they're closer to truth than being just dumb animals.

Ours is pretty good at warding off evil spirits. Now, if we could just work on her definition of evil spirits, life would be so much nicer! The rabbit in the back yard at 2am? Not an evil spirit, so knock stop the warding off!

Some dogs are scary smart (for dogs) but none of them can think at the level of abstraction where they imagine / create deities or really understand the fact of their own mortality for that matter.

We have 2 dogs, one of them is dumb as a post except in very narrow ways and the other is too smart for his own good. That one always reminds me of the meme where a dog is looking at the sunset and the caption reads, "What if I never figure out who's a good boy?" We literally have him on Prozac and Vallium or he would whimper all day. He's a rescue and it's very satisfying to us that he has improved greatly since we got him from the shelter at 1 year old, but he's a neurotic little guy just the same. But he doesn't have the gray matter to do anything with a god concept either.