what happens if I end up filing a complaint and they are responsible?
DMV will take action against the shop. This may include suspending or revoking the shop's registration and/or imposing fines. If you suffer a financial loss due to overcharging or unnecessary or poor quality repairs, the administrative law judge presiding at the hearing may offer the shop the option of paying restitution.
would I be correct in believing that the repair shop has to give me a paper listing what was done on my car?
Yes, it must give you a copy of a work order