I'm not sure about website but for date, period and segment it should be quite easy to implement for Piwik 3.0+. I did a very quick POC here: 83ce968

Pretty much the only thing that takes a bit of effort is to change the calendar and segment selector (and maybe website selector). So far the calendar selector never really closes as the page is just reloaded anyway. Instead we'd need to change the date selector, update the displayed date in the widget and tooltip, and update piwik.date and piwik.period JS variables if needed. It's all feasible though.

Would improve user experience quite a bit by removing these page reloads. The widgets / reports would just reload instantly.