I am trying to figure out if this is possible or not. My application is using the ArcGIS SDK to show a map. It is a standalone application, not a Desktop Add-in. I am required to be able to modify layer properties and symbology from within my application. So, instead of re-making the whole dialog (I'd like to invest my time in the added content, and not reinvent the wheel), I was wondering if there was a way I could just invoke the dialog that's available in the ArcMap application. I do not want to have the ArcGIS Desktop installed in the client's machine, though.
This is not possible unless you have ArcGIS Desktop installed. The layer property pages are found in the CartoUI assembly. The CartoUI assembly is not part of an Engine installation; it's a Desktop-only assembly: