What are the consequences of falsely and/or unnecessarily declaring an emergency?
My experience is it is whether ATC files a complaint against you. It has to be pretty serious for them to do that. It works both ways, pilots rarely report ATC mistakes and so there is a gentleman's agreement to not sweat the small stuff. I have had ATC try to kill me at least 4 or 5 times and I never saw the need to report it. I figured the controller already knew the seriousness of it and if a supervisor noticed it, more than enough bad things would happen to them. 

Just to be fair, I have screwed up seriously twice. I misunderstood instructions in flying over Vancouver International in  Canada and flew right in front of landing traffic. The second occurrence was when I was flying a twin. I got really busy after the controller unexpectedly routed me to a different runway during approach to Anchorage International, I switched tower frequencies for the new runway and forgot to get "cleared for landing" with the new tower frequency. In both cases the controllers very sternly told me what I had done wrong but never reported it.

In 1975 I had ATC file a complaint with my flight school when I was a student. I turned off the engine when it was obvious I had a 10-15min wait for takeoff. I later found out the FARs require at least one engine running on all ground control areas - no one had told me, Oops.

I have had two very serious problems in my 2500hrs of PIC. I lost an engine in a Cessna-337 (actually I was practicing engine out and one prop accumulator had leaked so the prop would not un-feather). ATC asked if I was declaring an emergency, I said no, and they declared one for me anyway. 

The second emergency was when a carb intake became obstructed by an air straightener on a Navion Bendix PS5 pressure carb. Right after taking off I lost about 50% power. I slowly circled back because of "..low fuel pressure.."  , and I said I needed a "precautionary landing". Once again they declared an emergency anyway and I landed safely.

Most pilots don't know the definition of an emergency - In my early years I didn't

One of the possible problems with getting in trouble with the FAA is not knowing what an emergency really is. (They are quite lenient and understand this gap in knowledge. For example I have never seen a FAA definition.)

In about 1985 I discussed my experiences over coffee with a 10,000hr Atlas B747 captain that had been an instructor. He listened attentively, then calmly ask me if I knew when to declare an emergency? I said "if I thought I might die!" He calmly instructed me... 


  An emergency is when "safety of flight is in doubt",
  
  OR when the aircraft is not able to perform as intended,
  
  OR when the aircraft might not be able to perform as intended. 
  
  It's that simple!!
  
  Also, a pilot is required to inform ATC (but not report a
  emergency) any time an aircraft is not able to climb at least 500fpm.


Note the emphasis on "might not be able to perform as intended". That may not sound like an emergency but that is how the FAA and ATC see it. And that is pretty low criteria - it gives the PIC lots of legitimate wiggle room.

Once I was told this, it made so much sense and I knew I could defend my actions as PIC in the future. It seems ATC likely has received better training than pilots on what an emergency is, even if the pilot is overly cautious to declare an emergency (as I use to be).

When the prop would not un-feather and I lost partial power, both aircraft would no longer perform "as intended" and I should have declared an emergency.