Matrices are a different type of object from vectors, so the transpose operation deserves a separate symbol $^T$.
An abstract vector space may have several different inner products, which must have distinct names, so $g$ is reasonable.
But all the others seem superfluous compared to the dot. The notation $a\cdot b$ correctly suggests bilinearity, that it acts like multiplication. It can also be used in an abstract (no inner product) space for the action of a covector: $\alpha(b)=\alpha\cdot b$