Can ANYBODY address whether there's more to it than "doesn't matter"?
I'm unclear:
Does the phone circuitry control the charge rate?
In short, yes.
Do better phones have better charging circuits--or do all just have a current-limiting resistor as the "charging circuit"?
Probably true, but no way to know for sure without knowing the circuitry inside each device.
I've been seeing high-current adapters putting out more than 2A--will the phone handle 2A or is the extra output wasted (BS marketing)?
Ohm's Law for DC circuits says that when voltage is applied across a given resistance, a given current will flow. Aside from using conducting wire that is too small for that current, and would add more resistance, and likely heat up and melt as a result, the amount of current that will flow is a constant tied to the internal resistance. The reason they make larger capacity charger wall warts is because some devices will pull more current than others. But if you have a device that draws 1 amp charging, it will do that even if you connect it to a 10 amp charger. So using a charger of higher amperage capacity than what the charging device is rated to use does nothing at all. And is usually safe. However, using a smaller capacity charger, such as half an amp (500 mA) will also work, BUT it will take longer to charge. Likely twice as long.
What if there was a regulated 5A charger--or I build one?
See above.
Does the phone cut off charging at 100% or continue to trickle charge?
If the device circuitry has a smart charger, it will taper off and stop charging at or near 100%, and hold that if left plugged in. If the charge starts to fall, it will resume charging, then stop again when fully charged. The S3 has such circuitry. I plug mine in all night every night, so it has a fully charged battery in the morning.
Or does the charger sense when the battery is full?
See above.
What current can a USB cable handle? Never any ratings on them...
It would be dependant upon how each one is constructed. Some have very fine wire, so can handle less current. Some have rather thick wire, which can handle more current. To really know that answer one would need to know the gauge of wire inside. There are charts where you could then match wire gauge to current. I do not know the current rating of the actual connectors typically used in a USB cable, but it probably is rated higher than the wire is rated in most cases.
Hope this helps. 
I'm not an EE but I once stayed at a Holiday Inn Express that was hosting an EE convention
;-)