Q1: Am I correct that I want a constant-current circuit to charge a supercap?
Q2: Am I correct that the charging voltage should start out at whatever the supercap's current voltage is, and then be dynamically raised high enough to maintain the constant-current feed as long as the voltage across the supercap never goes above the supercap's rated value?
Q3: Am I correct that when using a supercap as an energy source, I should always add a little resistor in series (or perhaps a current-limiter) to prevent the load from pulling too many mA?
Q1: Constant current with a max voltage limit yes.
Q2: You can't do anything
except start charging at the caps current voltage.
If you push massive current into the cap you may notice you're charging at 2V but when you switch it off the cap is 1.5V.
What you're seeing here is losses in the wire between PSU and cap. The cap was never at 2V, it was 1.5V with 0.5V loses in wiring.
So you need your charging system to be measuring voltage for the cut-off right at the terminals of the supercap.
Q3: Depends, it's true that a supercap will not hesitate to smoke your wiring in an instant if you short it out. So i recommend a fast blow
fuse with a good margin above normal current.
You only really need to add some extra series resistance on a supercap when using it for a circuit that expects a battery.
For example, in a car where the alternator expects to see a battery it will often stall the car at idle trying to charge much too fast because the supercap voltage-vs-capacity relationship is quite different to a lead acid battery.