What constitutes an "emergency" in flight procedures?
There is a really good explanation of emergency situation over at the SkyBrary:

An emergency situation is one in which the safety of the aircraft or of persons on board or on the ground is endangered for any reason.
An abnormal situation is one in which it is no longer possiible to continue the flight using normal procedures but the safety of the aircraft or persons on board or on the ground is not in danger.
Emergency or abnormal situations may develop as a result of one or more factors within or outside an aircraft, for example:

Fire on board the aircraft;
Aircraft component failure or malfunction (e.g. engine failure, landing gear malfunction or loss of pressurisation);
Shortage of fuel (or other essential consumable substance);
Flight crew uncertain of position;
Worsening weather;
Pilot incapacitation (e.g. as a result of illness);
Aircraft damage (e.g. as a result of collision, bird strike or extreme weather;
Illegal activity (e.g. bomb-threat, wilful damage or hi-jacking).


So in short, anything that puts the flight in danger can be considered an emergency.


Is ATC allowed to judge what is or is not an actual emergency, or must they prioritize any aircraft declaring one?

If the pilot declares emergency, ATC will provide assistance and support to the flight without judging whether it is an actual emergency or not. ATC cannot know in most cases, nor are they really required to analyse the situation to come to that conclusion. Their whole responsibility is to ensure the flight receives all required information and support to either end the abnormal/emergency situation or land at a suitable airport.

Does fault or blame matter at the time the emergency is declared? Will ATC de-prioritize a situation that is obviously the pilot's fault, like flying into the only cloud for 50 miles in any direction?

No, they will not de-prioritize based on the type of emergency. Even if a pilot flew VFR into IMC conditions, he is still in a life-threatening situation from which he needs to recover, so assigning blame or guilt is not helpful here, nor would that be the ATC's job. This is the Civil Aviation Authorities' responsibility, once the flight is back in safe conditions. ATC may file a report, though, but not during the emergency, but as an aftermath.

Is there anything you still can't do as PIC in an emergency situation? Apparently in an emergency you can do a lot of things that would get your license pulled in any other scenario, like violating separation minimums, entering airspace that's normally off-limits like R-zones and Class A space, running on instruments without a rating, etc. While these things are still dangerous, it is apparently the pilot's prerogative to resolve the emergency however he can. Is there anything a plane declaring an emergency still must do other than fly the plane, or conversely something he still cannot do like get too close to Air Force One?

Technically, once you have declared an emergency, you can do whatever is required to keep the flight safe and prepare for an emergency landing. If you need to enter otherwise controlled or restricted airspace, this will be made possible by ATC via coordination with adjacent units. Without knowing FAA regulations on this specifically, I would daresay that distress traffic could have priority over the Air Force One.

After declaring an emergency in flight, what should be done after the plane's back on the ground (hopefully safely)?

Unless your CAA or their respective investigative service is not already waiting for you, you should remain with your aircraft (unless being treated for medical issues and taken to a hospital) and wait for them to arrive. If in doubt, you can call the police and they will notify the relevant authorities.