I would like confirmation on the following: Is any metal box automatically grounded if it is supplied by an armored cable (without a ground wire) that is properly connected to the box and back to the panel? If a metal box downstream on the same circuit is only supplied by Romex, I believe to ground that box I must pigtail the ground wires to a lug on the box - correct?
Armored cable without a ground connected to a metal box is not a proper ground. The Romex downstream would have it's ground connected to a lug or screw to the metal box but it still wouldn't be properly grounded because of the armored cable. Voltage testers would show a ground but it wouldn't be a proper ground.
The answer to the first question is no, not all armor is suitable for an equipment ground. For example, MC cable with interlocking armor is used quite a bit but it's armor is not suitable for an equipment ground. It's almost always seen with a ground wire.
Other cables such as AC, which usually has a bonding wire welded to the armor, are suitable for grounding when used with connectors listed for grounding.
The answer to the second question is technically also no, because it asks if you "MUST" ground with a lug and pigtail. Yes you must bond the metal box to the equipment ground, but that's not the only way to accomplish it. You could put a green 10-32 screw into the threaded hole in the back of the box and wrap the bare wire in the Romex around that screw, that's probably the most common way. (By the way, no reason you couldn't use any You could use a grounding clip, once in a while that's expedient.