I am wondering whether or not self-interaction error and integer-discontinuity in Kohn-Sham density functionals, are related to each other?
Yes and no.
The exact functional should have a piecewise linear behavior in the number of electrons Phys. Rev. Lett. 49, 1691 (1982), but this is not true for common density functional approximations, and self-interaction errors do affect behavior at fractional electron numbers.
Now, if you apply the Perdew-Zunger self-interaction correction, then not only do you mitigate the one-electron self-interaction error (N.B. the correction is often problematic, see Adv. At., Mol., Opt. Phys. 64, 1 (2015) and J. Chem. Theory Comput. 12, 3195 (2016)) but also introduce step-wise behavior in the functional due to the added term; orbital energies in PZ-SIC are often better than in Kohn-Sham DFT. There's also been a lot of work on the so-called Koopmans' compliant functionals by Marzari et al.