SGC,
The beam spreads out over distance and is affected by a number of issues, mostly bending due to prox to heavy iron/steel concentrations such as bridges and buildings, hence "For most facilities, this really is linear but in some places it can be slightly S-shaped."
The glass in a building will also have an effect, all glass buildings will tend to reflect the signal that has reflected off of an aircraft, hence the aircraft may receive multiple bounces of the signal. hence "but I've experienced the autopilot attempting to lock onto a "false" LOC signal several times."
Following a large aircraft mass will upset the signal as well, perhaps giving multiple signals reflecting of various strengths, hence "Very steep (high sensitivity), and with lots of noise."
Construction density around airport facilities is a major concern, and the reason we are looking at shutting down the ILS/NDB and other beam type signals, and replacing the system with GBAS.