I really don't understand electronics at all, but I've read that GND is really just a point with zero voltage.
So couldn't you in theory stick a nail into a piece of plastic and use that as your GND?
No. Depending on which theory of electronics you want to go by, either electrons flow from positive to negative or negative to positive. But without delving deep into the whole electron flow theories Ground(GND) is not neccessarily a point without voltage. GND is usually the opposite of the side with voltage. Basically its where the voltage wants to go. One way to put it, its the negative side of the battery. And the voltage from the positive side is looking for a path to reach the negative side(the ground). So simply sticking a nail into a piece of plastic does not qualify as a ground since that nail is not in contact with the negative side of the battery.
In short, GND is not a point with zero voltage but usually the negative side of the battery/power supply etc.
In some instances not all grounds are the same so they cannot be connected together. I found this to be true on the Logitech Precision gamepad for the PC.
Also, can I hook the Perfect 360 GND to the Dreamcast controller button's GND? (eg. all the Dreamcast controller buttons and the P360 GND would connect to the same place on the Dreamcast controller)
Like I said before not all grounds are usually the same, and this is even more so for separate devices. There is a possibility they are the same and it will work. But in the instance that they do not share the same ground, and you connect them together its possible you can damage some electronic components rendering them useless.