Thanks for bumping this post. I’ve edited my original reply above to indicate that flashing / animated icons are possible with the release of the Themes and Styles feature which supports customization of the tile icon to add various animation effects. Power users can also take advantage of custom CSS for advanced visual effects.
Depends on the tile. A common approach is to use Variables for keeping track of things that are true/false or active/inactive. You can configure variable tiles to use the Hyperlink action and could even pair that with the special $.runRule() syntax to run a rule with whatever logic you desire.
The end result would be a single tile/variable showing your status and then clicking on it could run your desired action within a rule.