Yes, the basemap sets the coordinate system for the web map.
You can set the WMS service to be the basemap of your map (there is a checkbox when adding the WMS to the map that says "use as basemap"). If you do so, any data you add over it will need to conform to the coordinate system of the WMS service.
If you can configure your WMS service to support the web mercator aux sphere projection, I think it should work.
As far as mashing up layers from different sources, there is no magic here...all the data sources need to be able to display in the same coordinate system. If you use one of the provided basemaps, it's web mercator. Or you can use your own basemap.
Hope this helps,
Mike