Unfortunately, you don't seem to legitimate cause of action, as the insurance companies have very wide discretion when it comes to quoting policies. It would be one thing if your son disclosed all infractions and they quoted him $210 but then shortly thereafter upped it to $13,000/yr. However, your son did not give them an accurate history of his infractions, so it is understandable that they did not give him an inaccurate quote. Although you could certainly argue that they've previously been your insurer and they should know of his history, the fact of the matter is that it is his responsibility (not the insurer's) to provide an accurate history- this is why most insurance companies request the insured provide them with a Drivers Abstract from the Ministry of Transportation, so that they have a clear record of the insured's history.
The reason the insurance jumped so much is probably because most insurance companies allow for two infractions, and upon a third, the insured is moved into a different 'range' of the insurance. I agree that it is unfair to spring this huge price hike on you / your son with barely any notice, but I do not see any grounds for a legitimate lawsuit.
As for your latter question, you can try finding a lawyer on contingency (i.e. gets paid only if you win) but that may be difficult as most lawyer who work on contingency only do so for personal injury claims (e.g. slip & fall, motor vehicle accidents) where there possibility of payout is very high. That being said, you are welcome to search for such a lawyer by searching litigation lawyers (likely insurance litigation) who work on contingency. A quick google search should yield plenty of results.
Sorry if that is not the answer you were hoping for, but hopefully it helps give you a realistic sense of your situation.
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The insurance company should have pulled his abstract. However, they aren't legally obligated to do so. If the forms say $210, then this could support your argument that their lack of due diligence lead you to incur excessive costs, and also resulted in your relying upon inaccurate information. You could try and argue that they should be stopped (i.e. 'Estoppel') from reneging on their prior quote. Again though, this may be a difficult argument as their quote was based on your son saying he had 1 infraction when really he had 3.
With respect to your latter question, Small Claims Court is cost effective, as it only costs about $100 to commence an action, and you can represent yourself / needn't hire a lawyer. Small Claims Court would likely have authority to hear your matter assuming you are suing for an amount that does not exceed $25,000. So, if you represent yourself, then the total costs of Small Claims from start to finish should be less than $500, if that.
Hopefully that answers your question.
Yes, if you lose you would likely have to pay some of the Defendant's costs, possibly betwen $500 - $2000 depending on the circumstances.
As for your latter question, the matter has likely been resolved with all parties being released from liability. If not, you could try suing the driver (which would go to his insurance), but again since the matter has resolved already I'd be surpised if you had any success. Also, there is a 2 year statute of limitation running from the date of the accident within which the action must be commenced.
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