My first question is: Should I be testing like this, with only the ground and 5V line hooked up (12V line not hooked up), do power supplies tend to act differently when not all the lines are connected?
It depends on the power supply's design. The simplest structure is for it to have independent linear regulators, fed by independent transformer taps. In this case, the output regulation is totally independent. You can test 5V and 12V separately and they do not interact at all.
However, this design increases costs, since it requires more transformer windings.
Another design would be to have a single winding on the transformer, with a 12V regulator, and then the 5V regulator runs off the 12. The outputs are mostly independent, but the 12V supply needs to be oversized, and it is also inefficient (dropping 12V to 5V is a lot of waste heat). This would be a strange choice but could be reasonable if the 5V load is small in comparison to the 12V load.
A switching design is more efficient, and may be cheaper as well since a large transformer is not involved. But if they were pinching every penny, the supply may have been designed to only regulate one of the output rails, meaning that the other rails will not perform correctly if that primary rail is not sufficiently loaded.
My second question is: Is it common for power supplies to fail in this way, i.e. as they get older they fail to give out the correct voltage?
The most common failure of linear regulators is a short or an open, i.e. they let full unregulated voltage through to the output, or they don't output anything at all. Switching regulators most often suffer damaged capacitors causing noisy output and reduced current ability.
My Final question is: Would the length and thickness of the cable between the plug on the amiga power supply and my current load affect the result?
If you are measuring the voltage at the load, then it could be that the difference is being lost in the cable. It would be more correct to measure voltage at the power supply output.
In situations where the load is at the end of a long cable, power supplies sometimes have remote voltage sense facilities to compensate for cable losses. I haven't ever seen one like that used in a computer, though. In the absence of remote sense wires, the output voltage is regulated at the feedback node of the power supply, not at the load.