According to Rav Ari Marburger (http://www.jlaw.com/Articles/Arkaos%20V1.3.pdf) Maishiv Bhalachah 42 permits suing an insurance company.However, Rav Yitzchok Zilbershtein in Yeshurun 11 requires requires a Heter Arkaos.I was one a passenger in a car that was in an accident and I was allowed to file a claim through an attorney. The driver actually told me to do so so apparently one claim will not affect the premium (this was in Israel so it may be different than in other countries and rate policies might differ from company to company).
As for winning in a din Tora where a child caused the damage (@Health) a bet din can order the parent to pay as a peshara, or even din if the parent was negligent in supervising the child (Rabbi Moshe Sternbuch, Teshuvot VeHanhagot 3,477) or gave the child the instrument of damage (Baba Kama 9b and Shulchan Aruch Choshen Mishpat 396,6).