Yes, quite frankly it is possible to find a same gene if you're in the same class of species, but finding the protein....I believe that's impossible because in every type of gene, you have the same proteins that make you function the same way. Without them you wouldn't be able to function properly.
If I found the same gene in all organisms that I've tested, I would be intrigued because that would be a giant step in evolution. My reason for this answer is because if you have the same gene that would technically mean we all specifically came from the same species of animals.
No, that's not true because other characteristics would eventually help us in many things, studies would help us get our brain much stronger and the intelligence level would be extraordinary.