  The Ising spin glass in two dimensions exhibits rich behavior with subtle
differences in the scaling for different coupling distributions. We use
recently developed mappings to graph-theoretic problems together with highly
efficient implementations of combinatorial optimization algorithms to determine
exact ground states for systems on square lattices with up to $10\,000\times
10\,000$ spins. While these mappings only work for planar graphs, for example
for systems with periodic boundary conditions in at most one direction, we
suggest here an iterative windowing technique that allows one to determine
ground states for fully periodic samples up to sizes similar to those for the
open-periodic case. Based on these techniques, a large number of disorder
samples are used together with a careful finite-size scaling analysis to
determine the stiffness exponents and domain-wall fractal dimensions with
unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and
$d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a
new uniform sampling algorithm allows us to study the domain-wall fractal
dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate
the distributions of ground-state energies, of domain-wall energies, and
domain-wall lengths.
