Q: Brakes give out while driving

Hi there. It sounds like you have a pretty good idea of what the brakes need. If the rear brakes need to be replaced then that should be done prior to replacing anything else as you do not want to replace any unneeded parts. If the rear wheel cylinder cannot be removed from the backing plate and if it needs to be replaced then the backing plate needs to be replaced with it. Once the rear brakes are repaired the remainder of the braking system can be checked. If the brakes still do not function and if the pedal is going to the floor then it may need a master cylinder. A mechanic typically pressure tests the system at that point to identify the issue. If you need assistance, I would have a brake safety inspection to determine exactly what repairs should be done.