Is the Runway Visual Range (RVR) communicated to the pilots?
Pilots get the RVR values either from weather reports or directly from ATC. RVR also shows up on ATIS reports when visibility drops to less than 1 mile. ATC will have a realtime readout of RVR and can communicate these values to pilots at any time by radio. 

ATC RVR display



Source: http://mtechsystems.com

The CAT III approach should only be commenced if the ceiling and RVR values meet the minimum for the approach in use and for the current aircraft equipment status. Some airlines will have their own minimums which can differ from the published CAT III minimums. 

Pilots will often ask for the latest ceiling and RVR values while doing the approach. ATC will also announce any changes or reduction in ceiling or RVR values while aircraft are on approach. If the new values are below limits, the crew should commence a go-around. 

A go-around can be done from any altitude and the wheels can touch the runway during the go-around. Low atlitude, "Low Energy" go-arounds are practiced during re-current Simulator training.