YES — Your compulsory third-party insurance remains valid if you travel to or temporarily stay in another EU country. The "Green Card" provided by your insurer — automatically or on request depending on your country of insurance — will make it easier for you to claim if you have an accident abroad. Other cover for damages to your car or to yourself (fire, theft, etc., included in comprehensive insurance) depends on your particular contract, so you should check with your insurer.
Because prices are still influenced a lot by local conditions in each country.
For example, compensation amounts for victims, civil liability regimes and the cost of vehicle repairs still vary widely between countries.