I wasn't going to chime in, but after reading all the replies I decided it was best I did.
First and foremost, different insurance companies will do different things - that's why there's not ONE insurance company, nor government funded.
If you have liability only, there's a good chance your insurance company isn't going to do jack squat for you, except post a claim to the other company based on what you said. The agent most likely won't pursue any compensation for you, nor make a single phone call on your behalf. This is YOUR job.
Even if you have FULL coverage, you'd be an idiot to not take it into your own hands.
Now, it's hard to determine the level of damage done to your car based on your description, but there are rights you have --- however they change from state to state.
In *MOST* states, you have the right to choose where you want your car repaired. Do your research, and find a good one. I can tell you from experience, the dealership is a good cop-out, but most likely could care less about you. I do not advise going through them.
INSIST ON RETAIL PARTS if there's any damage to components. The body shop is most likely at the mercy of the insurance company, and will NOT fight for you - that is NOT their job!!!!! They are there NOT to mediate, but rather just do repairs based on what was approved. No different than if you went in with an axle and wanted it replaced, that's how they'll treat the insurance claim; insurance guy says "use this $50 one instead of the OEM $200 one" and the body shop will say "Yessir, right away sir."
Insurance companies are going to attempt to use OEM replacments, in order to reduce their costs. Pardon the language, but bend them over and give it to them where it hurts. I have fought this fight so often with my aftermarket suspension, intake, etc etc.
As for compensation for the child seats, I'm green on this one. I don't have children, but I do know that if something was damaged internally such as seat belts, air bags, etc., it is absolutely mandatory for them to fix them. It's a safety harness afterall!!! You should find your receipt (hope you still have it!!!) to prove the cost of the unit, or again they'll try to get you the walmart special.
I have once been compensated $1000 for "convenience matters." The story went like this: I fought and fought an insurance company to pay for my car and my medical bills, all the while the agent said "no no no"... I called his supervisor who stated if the agent deemed it as unnecessary, there was nothing for him to do. Having told him that he was useless as a being, I found his regional manager's number, who again stated the same thing. I had enough and went to the Texas Board of Insurance (TDI) (which btw every state should have, I believe it's required by federal law - but I might be wrong), who filed a formal complaint against the company. After 2 weeks of having submitted my final medical bills, and having paid out of my own pocket for EVERYTHING, the insurance company phoned me to say "we'll pay for your bills, plus another $1000 for inconveniencing you." I took it and didn't say a word.
Insurance companies should only raise your premium amount if you are at fault of a moving violation (running a red light/stop sign, driving intoxicated, hitting another car). Since you are not at fault, I'd find it REALLY difficult to believe they'd ever raise your premium.
BTW, YOU SHOULD INVOLVE YOUR INSURANCE COMPANY ALWAYS!!! You're naive to think they don't have it on file as it is. The other insurance company has looked up your license plate or vin, or even used information swapped directly from the other insured motorist. Your information is on file at your insurance company indicating you were in an accident - please don't think otherwise.
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EDIT:
I read some more of what you said, and the bumper is repairable... What are they doing? Fusing it or replacing it?
What are the total estimated damages accrued from the accident?