I worked with Chrome extensions which have so called background page - an html page that is loaded in background once per browser window. You can store there some javascript variables, can access extension's own localstorage, can communicate back and force with content scripts (scripts injected to pages).
Is there anything similar in Firefox and how do I use it for the tasks listed above?
If you are using the (relatively) new Add-On SDK, then the main javascript file residing in your lib directory is equivalent of a Chrome extension's background page - a persistent script that runs in the background and spawns/creates/inserts panels, widgets and content scripts.
Regarding your specific asks:
1. localStorage: Add-Ons in Firefox cannot access localStorage directly. However, you can use simple-storage for storing data similar to localStorage.
2. Communication with content-script: Add-ons can communicate with content scripts using port or postMessage.
From the point of view of a traditional Firefox extension, the browser itself is just another window containing a document, although this is a XUL document rather than an HTML document. So you can store per-window variables, although you have to be careful not to overwrite other extension variables, which usually means declaring a top-level object and adding all your variables as properties of that object.
Sharing variables between windows used to be a little harder but fortunately JavaScript modules solve that problem in simple cases (primitive types).
Extensions can communicate with content scripts although there are some wrappers in place to prevent you from accidentally doing something silly.
