What can I expect from an insurance company after my car accident?

You can expect insurance companies to lie to you. They are in the business of lying, misleading, and denying claims. They want to take and take but never give back. It's their business model. Take in as much money as possible and pay out a fraction of that if they can get away with it. And they count on injured parties settling cheap, so expect the at fault person's insurance carrier to try and trick you and force you into a quick settlement. That's how they make money.