Hi Robert,
theoretical this works fine, but I have some timing problems.
First of all i had a look at the documentation 'Communication to app container' and the 'Dijit Lifecycle' to find out which is the correct method to place the code for minimizing the widget with the PanelManager.
I placed the code in the 'startup' method and it did work with Firefox but not with Internet Explorer.
To make it work in the Internet Explorer i had to use a timeout:
var panel = PanelManager.getInstance().getPanelById(this.id + '_panel');
setTimeout(function() {
panel.onTitleClick();
},500);
Using a timout is not the best solution 
I've tested the widget on some mobile devices and the "openAtStart" does not trigger the widget to open at start at all. So i guess for responsive design it does not open cause the display is too small. At the beginning i was only previewing it in the Web AppBuilder and there is a different behavior - as described above.
- Stefan