Oh wow, I feel for you - TN is not pleasant in the least. I can't say that I have this but I do have neuralgic pains elsewhere and certainly I can remember the godawful neuralgia I had with my impacted wisdom teeth putting pressure on the nerves at the bottom - the shooting pain referred through all of the teeth and bottom and top jaw on the affected side (occasionally both together if my body really wanted to make me miserable), up my face, down my ear and throat, made me salivate furiously, made my eye and nose run.... does this sound at all familiar? I know it isn't the same, by a long shot, because TN is reputed to be one of the (if not the) worst pains possible to experience.
As to whether dental treatment might trigger it off - I have to say perhaps. I imagine it would depend on what other things tend to trigger it for you - I know for some sufferers brushing their teeth can set it off, shaving, even just cold air. I don't believe that the trigeminal nerve, itself, innervates the teeth and - as such, it's not going to be irritated by any work on the teeth directly.... but I know that it is involved in the muscles that you use to chew, so I don't know if having to open your mouth for a while (or repeatedly open and close your mouth) might be an issue.
As for painkillers - I was under the impression that standard painkillers are all but useless for TN - and nerve pain in general. I know for sure that even the morphine that I am currently taking is all but useless for the nerve pain I have with my other health issues (I currently have the joy of low dose tricyclic antidepressants and on and off various anticonvulsants to try and manage that part of my pain). What is it that you normally need to manage the pain? Do you get any relief from applying either heat or cold to the trigger areas when the pain starts? If so, I'm sure it would be possible to keep say one of the chemical cool-packs in the surgery in case the pain is triggered.
I just googled quickly and found a post from a dental nurse who does say for one person that dental work can make it worse... but as to how many people have problems with dentistry and how bad, I can't say, I'm afraid. Could you call your dentist and ask them? Or perhaps ask the resident dentists for advice?
I wish I had some better advice for you - or at least some personal experience that was more use.

You really do have my sympathy though. Neuralgia is the kind of pain that is (without doubt) worse than any other kind of pain - especially when you have the numbness, tingling, electric shock sensations, weird touch and odd hot or cold sensations along with it. I hope you find a way to manage or control it soon.