When an appraisal is made on a repairable car, any needed replacement of tires and batteries is usually done on a prorated basis. That is, the insurance company will only pay for the unused portion of that part. The paint color of repaired parts should be made to reasonably match the rest of the vehicle.
After an agreed repair price has been reached either from an appraisal or estimates from a body shop, the vehicle can generally be repaired at the body shop of your choice, provided the shop accepts the agreed-upon repair estimate, which they normally do. If it is determined during the repairs that the original estimate is too low, a supplemental appraisal should be made. The repair shop should call the adjuster back to do a supplemental appraisal.
Please carefully inspect and test drive the vehicle after the repairs have been made. The manager or other employee of the shop can usually go on the test drive with you. Make sure that the vehicle is performing up to pre-accident standards before you accept it back from the shop. It is easier to have any problem corrected immediately rather than at a later date.