Why is the concentration halved at the intersection of Hg²⁺/Hg₂²⁺?
The total concentration of all ions is $c = \pu{0.10 mol L-1}$. The conservation equation of element mercury requires that
$$\left[\ce{Hg^2+}\right] = \left[\ce{Hg2^2+}\right] = \frac{c}{2}.$$

Generally, it's a convention to make the concentration of ions at the border equal. On the other hand, matter conservation implies that the concentration of each ion is $\frac{c}{2}$.