What a beautiful article, Umair. Yes, you're right, "I feel, therefore I am" is the way to go - arguably even viruses might feel. Certainly bacteria do.
However I don't think that your (admittedly adorable) puppy is "Confronting the strange and terrible fact of his own existence" when he sits in the park and stares into space. Why would he worry about something so boring? He's probably trying to get a visual fix on something he's just smelled, or looking for a friend who he can smell is coming to the park or wondering how he can get his strange apedog packleader to give him another treat.
He definitely doesn't ask "Why am I here, dad? Who made me?"
He's here because he's here. Now, this moment is enough for him. He's happy when he's staring into space and joyful when he's running around with his friends. If he was sad for any reason, you'd know it, even though you're human - I once took my lovely yellow labrador to the vet because, as I said to the vet, "He has sad ears." And he did. And it turned out he had a temperature and mild infection.
This is an even more profound matter than the nonsense of "Cogito ergo sum." It's what the Buddha was pursuing and found; it's what Marx was chasing in the wrong place. All animals naturally have this ability. If this moment is ok, they're happy. If they're hungry or in pain, the moment isn't ok and they try and change it. Most animals simply don't know how to worry - though some dogs with separation anxiety seem to have learned.
We don't even have a word for it, except in the concept of the Garden of Eden.