Must the car won through coupons—which was purchased with shared funds to which I did not contribute—be sold and its price distributed among colleagues, or does it belong to me?

If you win the car from coupons, then the owners of the money are the ones who deserve it, or they deserve its price if it is sold. The registration of the questioner's name on the coupon does not make him the owner of the car or a partner in it, unless the owners of the money give him permission. If one of them gives permission, then that is considered a gift from him to the questioner, and the rulings of gifting apply to it.