In reading through the more informal literature (Wikipedia, FAA PDFs, Av.SE), I often come across a caveat to standard flight rules and procedures: "unless an emergency exists". The intention is obvious; ATC is going to prioritize helping an aircraft that encounters a situation that simply cannot be ignored. However, most such resources don't go into specifics about what kind of situations constitute an "emergency". There are obvious emergency scenarios, like a "dead stick", and there are obvious non-emergencies like "I have to be on the ground by 5:00 to beat traffic home". Then there are grey areas like "I fought a headwind the whole way here, I'm low on gas and now my home field is IFR and won't grant me SVFR. I might make it to my alternate even further upwind that's clear, but I really think it's best to set down here even if I can't see the far end of the runway".
So, the questions:
What kinds of things constitute a genuine emergency, and what will get a pilot laughed at by ATC (or worse)? Where's the line?
Is ATC allowed to judge what is or is not an actual emergency, or must they prioritize any aircraft declaring one?
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?
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?
After declaring an emergency in flight, what should be done after the plane's back on the ground (hopefully safely)?