First of all let me say how happy I am that you added an x-ray, makes things much easier to discuss!
While its not the best xray in the world, we can see a few things.
1. Dark area below the tooth is the infection in the bone.
2. Dark area below the contact point looks like you have decay. There may be no hole in the top surface at all. Decay below the contact point is the second most common spot. So there is the original source (your hole).
(Its hard to see as the teeth are overlapped in the picture, personally I would have had my assistant retake it)
Now, lets go through the history a little to help explain. The original pain is caused by the bacteria attacking the nerve (sharp stabbing pain), which can last for a while. The pain disappears when the nerve dies. (pain that becomes no pain without cause is never a good sign). This is further supported by the abcess that formed. The bacteria grows constantly and finds the least path of resistance to drain. Pressure created by the bacterial growth causes its own pain (dull throbbing pain) and will mostly go away when a fistual appears because the bacteria can drain (pus) and the pressure is released.
An emergency case?
Most dentists would only consider it an emergency if:
1) You are in pain that prevents you from sleeping
2) Your infection hasnt drained and can spread into facial spaces
However, I would consider it an urgent case and I am surprised your dentist has not already given you options to take care of it immediately.
Based on what I see, I would say you have mesial decay with resultant nerve death that needs to be taken care of. I would suggest either:
1) root canal and crown
2) Extraction and implant
While your dentist is correct that there is not a 100% chance of #1 being successful, I don't think its as low as 50%, but I can look that up for you if you want. #2 option has a 95-97% success rate dependant on a clear medical history.
I hate saying bad things about other dentists, however I would not have waited on a case like yours. If he can't give you immediate treatment, I would find another dentist. Since I believe the problem originates from a cavity, you only need a general dentist, I see no bone loss or calculus buildup to necessitate a periodontist for option #1.
You can expect to be sent to a specialist for the following reasons.
Depending on the dentist, you may be sent to an endodontist for the root canal (many GPs don't like doing root canals in their office). An implant case would probably involve either an oral surgeon or periodontist, but you need to start with a GP to plan out the final results first.
If you have any other questions, please ask either here or in PM. Good luck with your case!