Is there a good survey on applications of Kirchhoff's circuit laws to graph theory or/and discrete geometry?
Possibly of help:
Pavlo Pylyavskyy, UMN Math 8680 Spring 2017, scribe notes (by students). (Scroll down to the attachments section for the scribe notes. The papers are less closely related to the electrical networks theme.)
The canonical reference on all thinks Kirckhoffian is
Doyle, Peter G.; Snell, J.Laurie, Random walks and electric networks, The Carus Mathematical Monographs, 22. Washington, D. C.: The Mathematical Association of America. Distr. by John Wiley \& Sons, New York etc. XIII, 159 p. \sterling 22.00 (1984). ZBL0583.60065.
I really liked the discussion of electrical circuits in the recent book "Probability on Trees and Networks" by Lyons and Peres. Chapters 2, 4 and 9 seem the most relevant to what you want.