Your zoomlevel set is confused, if you only want to render level 3 to 13, you can just set the style from 3 to 13, I think the code like this should works:
ShapeFileFeatureLayer worldLayer = new ShapeFileFeatureLayer(MapPath("~/SampleData/world/cntry02.shp"));
worldLayer.ZoomLevelSet.ZoomLevel03.DefaultAreaStyle = AreaStyles.CreateSimpleAreaStyle(GeoColor.SimpleColors.Transparent, GeoColor.FromArgb(100, GeoColor.SimpleColors.Green));
worldLayer.ZoomLevelSet.ZoomLevel03.ApplyUntilZoomLevel = ApplyUntilZoomLevel.Level13;
If you want to make user cannot zoom to level 1, 2, 14 to 20. I think you can set the custom zoomlevelset, which only leave 3 to 13 levels and remove all the other levels.