When ∆Suniv = zero, does the concentration of reactant equal the concentration of product?
All thermodynamics boils down to looking at $\Delta S_\mathrm{univ}$. $\Delta S_\mathrm{univ}$ is the sum of $\Delta S$ for your system and $\Delta S$ for everything else. We measure $\Delta S$ for the system directly and we measure $\Delta S$ for everything else via $-\frac{\Delta H}{T}$. When these two are equal, notice that $\Delta G = \Delta H - T\Delta S = 0$. Then given $\Delta G = -RT \ln K$, we conclude that $K = 1$.

The reaction favors reactants as much as products, but that specific balance depends on the reaction and its coefficients, i.e., the equilibrium expression could have powers. Consider:
$$\ce{A2 -&gt; 2A}$$

The concentrations of product and reactant are not equal at equilibrium for $K = 1$.

Also, it does not mean that a reaction does not occur. You need to compare the reaction quotient $Q$ against $K$ to determine if there is a reaction or if we are actually at equilibrium.