Do non-functional (‘junk’) protein sequences exist?
This question is unanswerable as, if a protein exists as a physical entity in a cell it is possible to demonstrate it has a functional or structural role, but it is logically impossible to demonstrate it has no such role. The best one can say is that “it has no known function (role)”.

That said, in the course of evolution following gene duplication and divergence it is theoretically possible that on the way to acquiring a new functional role there is a stage at which a protein has no such role. One might be able to identify such proteins in a gene family like the globins by examining a range of species, as has been done for the globin genes. I am not aware of any such studies.