Was the Universe's entropy equal to zero at the Big Bang? Is zero-entropy state unique?

Whether entropy was zero at the Big Bang or not is very much an open question of physics, in big part due to the fact that we do not yet have a good enough understanding of physics at high energies and high gravitational fields.

But for the zero entropy state this is a bit easier to answer and the answer does depend on laws of physics. Zero entropy state basically depends on how many completely distinguishable states the laws of physics allow. The universe is in a zero entropy state precisely when it is in a single state and it can be known which state it is in. In many situations there are infinitely many different zero entropy states. So the zero entropy state at the beginning of the universe is unique if and only if the laws of physics at that time require that there is a single state in which the universe can be found. Whether they do require that or not is a very big question in physics which everybody would like to know the answer to.