"A shred" is a small scrap. Think of the verb "to shred", and the pieces that you get from shredding something are called shreds.
The incident took a piece of the writer's ego, ie. she had a very small experience of humiliation. She felt stupid because she thought her phone was buzzing, only to discover it wasn't.
For the second question, I understand it to mean 13% of the original 68%, which works out to about 9% from the overall survey.