"Gossiped away" means the same thing as "gossiped", essentially. The "away" emphasizes that the gossiping is uninterrupted.
The cat is staring at mister dursley, she's returning his stare, giving it back to him, essentially. This is a pretty common usage. For example: "He kissed me, so I kissed him back."
Yes, the "nerve of him" is an idiom. It's exclamatory in tone, and expresses outrage at a person's behavior.
"Single" here means that he couldn't see any collecting tin at all, not even a single one.