1) searching a word in layer list is so awesome. I love it. It works fine if you type a word in. For example on my site try 'Schools'. However... if you typed School, but then add an 's' after waiting a second, it collapses the service. t should not collapse the service. It seems to simply be a CSS bug or issue. Users will have to know to expand the Map Layers service (that is its name). And they won't. Trust me. Sidenote: users do NOT understand this interface of the layer list. It needs partial checkboxes if only some of the sublayers in a group are on. And it needs to auto turn on a layer even if the parent is off.
2) You should add a 'title' attribute explaining to expand layers by clicking their name or arrow, and also that if layers are grey the map needs to be zoomed. I put an HTML in my widget.html. I recommend Esri do this and make the left dropdown arrow larger and make an-hover or title attribute saying Click Here to Expand. See my site as an example: SAGIS Property Map Viewer