What an insurance company requires of a repair shop with regards to car parts used for the repair of a vehicle after an accident differs according to state laws and then the terms of an insurance policy. Also it can make a difference if you are making a first party claim (one through your own insurance company) or a third party claim (one made through the at-fault party's insurance provider).
The Ohio Department of Insurance states in their consumer guide that if you have made a claim under your own policy you have the right to choose which shop will repair your car, unless this right is limited by your policy. You are entitled to a prompt and good-faith settlement offer based on the terms of your policy and to negotiate with the insurance company's adjuster. You have the right to reject any settlement offer that you find unacceptable and use any arbitration procedures provided in your policy.
The Ohio insurance regulator also notes that the selection of crash parts is determined by availability, cost and quality. When a vehicle is damaged, the insurer has an obligation to provide fair and timely claims service. The industry is obliged to make sure replacement parts are of “like kind and quality” also referred to as LKQ.
So in Ohio you can normally have your car repaired wherever you choose. However, regardless of what shop you choose, the adjuster will base your claim payment on market price for the repairs and pay the local average rates for parts and labor. You may pay extra if your shop charges more or if you want OEM parts and the insurance company wants LKQ parts used instead.
An insurance company is not typically going to tell a repair shop where to get parts from but may state what type of parts (OEM or LKQ) they will pay for. If you want new parts and they are paying for used parts you would have to pay the difference. You can contact the OH Insurance Department for more consumer advice. You can explain your particular situation and they can tell you what Ohio insurance law has to say about it.
As for who the check is made out to, you will need to ask the insurance company involved who it will be made out to. Depending upon your situation the check may be made out to you, you and the repair shop or directly to the repair shop. Also if your car is financed the check may be made out to you and the finance company. The lien holder' name would be on the check because your car is collateral for your loan. The bank (or whoever is financing the car) has an interest in making sure the money is used to repair the car and not for anything else.