2) From what I learned about traffic laws, "Turn right on red" car should yield to cars going straight. You did not hit the back of their car is a valid proof that they are at fault. Also, check the intersection to see if there is sign saying "no turn on red".
3) Call your insurance company. Tell them what happened.
4) It is their fault. Their insurance company should pay you. If you have full coverage insurance for your car, you will get paid no matter what.
5) Very Important: As you have kids in your car and your car is in an accident, you need to CHANGE car seats. Insurance will cover the cost.
6) If you think there is personal injury, go to a hospital to check out.
7) As your kids are in the accident, keep all the documents for at least 18 years.