1. If a voltage regulator, like the 7805, has the correct input voltage, but incorrect voltages at all the other pins, is it by definition faulty? Or, can problems with caps or load cause this?
2. Am I correct in thinking some (power supply) voltage regulators handle only ac and some only dc? Do some do both?
"at all other pins"? there are only three pins on a 7805 - normally one should be grounded, one input, and one output. If your input is correct and the ground pin is in fact grounded, and the output is not 5v, then the regulator MAY be faulty, but based on your confusing question it sounds like there could be any number of other issues that aren't the regulator's fault...