  The point process of vertices of an iteration infinitely divisible or more
specifically of an iteration stable random tessellation in the Euclidean plane
is considered. We explicitly determine its covariance measure and its
pair-correlation function as well as the cross-covariance measure and the
cross-pair-correlation function of the vertex point process and the random
length measure in general non-stationary regime, and we specialize it to the
stationary and isotropic setting. Exact formulas are given for vertex count
variances in compact and convex sampling windows and asymptotic relations are
derived. Our results are then compared with those for a Poisson line
tessellation having the same length density parameter. Moreover, a functional
central limit theorem for the joint process of suitably rescaled total edge
count and edge length is established with the process $(\xi,t\xi)$, $t>0,$
arising in the limit, where $\xi$ is a centered Gaussian variable with
explicitly known variance.
