Is it possible to keep the web-widget history / session when reloading or changing the web page ?
Yes that's possible, but you will need to take care of this yourself. It's currently not in the core of the web-widget code.
So you would probably want to store the text in the local storage or something like this.
Here's the web-widget code, if you want to modify it: https://github.com/botman/web-widget