1. That's not a tap washer, it is a silicone rubber ring. With 1/4 turn taps it is used to press the two ceramic valve segments together.
2. They can compress over time, not exert enough force, and allow water to seep between the segments.
3. You need to find a cartridge of the correct type and replace the whole cartridge. I'd recommend doing hot and cold at the same time. If one side has gone, it's quite likely the other will follow soon after.