1. Yes, there is a bug displaying some layers on mobile devices. This will be addressed when the apps update.
2. We do plan to enhance the My Content display to be able to show thumbnails. Also to be able to more easily open items like a map without having to view its details first. 
3. Yes, it would be nice to reorder layers by dragging. Explorer Online has the advantage of the Microsoft Silverlight plugin, which provides additional capabilities. There are also other technical issues that make this challenging because certain layers (those drawn by the browser) can't be displayed below layers drawn by a server. It's easy to gray out buttons when you can't move a layer any further, but if you're dragging, we'd need a different visual way to let you know.
Both the arcgis.com website and ArcGIS Explorer Online are client applications that access your maps and data stored in ArcGIS Online. They were developed simultaneously. Silverlight provided additional user interface capabilities, but some users didn't want to install a browser plugin to use it. In general, web development appears to be trending away from utilizing plugins, thus we are focusing our efforts on enhancing the arcgis.com client.
Thanks for your feedback. We'll get the bugs fixed, so please continue to let us know if you find any more.
Mike