Contact Inhibition of Cell Division: Signaling Pathway
How does the mitotic entry machinery roughly respond to cell volume?


This is a broad question but luckily there is one article that addresses this precise question in detail. However, it is very likely that more discoveries are made on this topic in future.

Neurohr et al. (2019) have extensively studied the effect of cell volume on mitotic re-entry and cell cycle progression. They used Saccharamyces cerevesiae with temperature sensitive alleles of CDC28 to arrest the cells in G1 phase. At this stage high glucose concentration (2%) led to increase in cell volume compared to that of starved (0.1% glucose) or protein synthesis inhibited (2% glucose + cycloheximide) cells. Using this model they find that cytoplasmic dilution causes impairment of transcriptional regulation during cell cycle progression. This is because the transcription factors do not scale with cell size. 

Read the paper for more details. 




  What's a relatively straightforward downstream pathway (core
  elements), starting at the adherens junctions, that illustrates how
  cell-cell signalling gets the STOP signal to the mitotic entry
  machinery?


This is again a broad question if you expect all details. However, google scholar search on "contact inhibition pathway" suggests that the main downstream pathway is the Hippo signalling pathway. 

Top hits on this topic:


Zeng, Qi, and Wanjin Hong. "The emerging role of the hippo pathway in cell contact inhibition, organ size control, and cancer development in mammals." Cancer cell 13.3 (2008): 188-192.
Kim, Nam-Gyun, et al. "E-cadherin mediates contact inhibition of proliferation through Hippo signaling-pathway components." PNAS 108.29 (2011): 11930-11935.
Gumbiner, Barry M., and Nam-Gyun Kim. "The Hippo-YAP signaling pathway and contact inhibition of growth." J Cell Sci 127.4 (2014): 709-717.