Closes https://github.com/microsoft/playwright/issues/32159. I
originally set out to enable Strict Mode for our React UI, but found a
way better thing: Enabling the lint rules we had already installed!
`eslint-plugin-react` is already in of our `package.json`, and this PR
enables it and fixes some of the reported issues. Most of them are
around the `key` prop which is mostly about performance, but there's
also fixes for misspelled `data-testid` props.
* Column widths are now stored on in the NetworkPanel context, this way
they are not reset after selecting an empty range (and changing position
of the NetworkGridView in the component tree).
* Column widths values are now preserved if column set changes (e.g.
selecting entries from a single context and then from multiple
contexts).
