What are the safest procedures for treating root canals?

Rubber dam. Use a rubber dam- it is the standard of care to make sure you do not aspirate any of the files. Your dentist is a professional and will do a great job and you will feel better. Any concerns speak with dentist prior to the procedure and have a treatment plan how you will restore your tooth after the root canal is complete.
Safe Procedure? Have faith and let your dentist or endodontist take the responsibility.
See a specialist. See the endodontist, the specialist in root canal treatment. They will use the latest and most proven techniques. The rubber dam is the standard of care -- very safe and protective. Many use the operating microscope -- excellent. Good luck. Expect to be pleasantly surprised.