Thank you Jayanta.
Let's start with (2), this was very helpful. Even when not using a basemap, but several layers of different coordinate systems, projecting some feature classes to use the coordinate system set for the data frame does significantly improves experience.
Regarding (1), I guess I've tried this without explicitly setting up a connection. I created a new empty data frame and loaded OpenStreetMaps. But I tired your procedure as well. In both cases, panning and zooming are nowhere nearly as smooth as with ArcGIS online, which is incredibly smooth.
EDIT: I get a smoother pan and zoom when I click the "Preview" tab in ArcCatalog as opposed to dragging the map from ArcCatalog to ArcMap.