I don't see a cause for concern. He wants to review the correspondence and records to advice you properly. However, it is up to you if you want to leave documents with an attorney you have not retained. Please remember that any communication with this attorney will be confidential between the two of you even if he is not retained.
You can negotiate and pay them what they accept. They have not filed a statutory lien so technically they are not allowed any portion of your settlement by statute. However, you owe them for the services they provided. So negotiate as much as you can to pay as little as you can.
You don't need to hire an attorney in your county. Any experienced attorney locally can help you. Your bodily injury case can not settle until you are done with your medical treatment. Your attorney can guide you through the process.
If you and your passenger have sustained injuries, please obtain medical treatment right away. Contact a personal injury attorney for a free consultation. Make sure to file claims with both insurance companies and submit an SR1 to the DMV if your property damage is over $750 or anyone has sustained injuries in this accident.