If I wanted to control several ICs that exceed the current rating of an Arduino 5V supply pin with an external DC regulator (which also would power the Arduino itself), should the ground planes of the DC regulator be be connected to the Arduino GND pins?
What we call a Voltage of 5V is really a Voltage Potential of 5V. The difference is that a Voltage Potential is the difference between one Voltage Level and another Voltage Level. 5V can be +3V to -2V, +12V to +7V, or 5V to 0V. They are all the same. Ground is also a relative term. It's the lower voltage level used as a reference to the higher voltage level (or vis-versa when using negative voltages).
Since the voltages are relative, two arbitrary voltage sources could be different. To make them work together, you tie the ground reference voltages together, so that all the circuits have the same reference.