Im sure this answers your question. And for your update, Yes you can use localstorage to store info on the client computer. Hope this helps :)
Well to track how many installs of your extension, you can add the
Google Analytics ID to your extension in your management area for you
Chrome Extension Gallery. That will keep track of the installs and
updates. Google analytics even has tracking usage, so you can add
tracking information based on a specific metric (clicking, logging)
etc.
If you meant for each "user" tracking, no, extensions don't identify
each individual user, the only thing that is unique is the extension
itself with its extension id. You can have other means of identifying
a user uniquely through a third party service, such as OAuth, OpenID,
or some custom stuff. Then you can associate your authentication to
something locally