I am having some confusion over the Metropolis algorithm. Let $g(x|y)$ be our proposal distribution for the algorithm. For the Metropolis, $g$ must be symmetric (from Wikipedia). In the discrete case, we can think of this as a stochastic matrix. I have also read, that
Transition matrices that are symmetric $P(i,j)=P(j,i)$ always have detailed balance. In these cases, a uniform distribution over the states is an equilibrium distribution.
I assume this is because symmetric matrices are doubly stochastic.
Could someone please clarify the ties between the Metropolis algorithm, Metropolis-Hastings, detailed balance, symmetry and the uniform distribution? More specifically, when does the Metropolis algorithm yield a uniform distribution (or why is it not always uniform since our $g$ is symmetric)?