Is there a way to flip the direction of all selected lines at once?
The Editing toolbox in ArcToolbox has a tool called Flip Line, which will reverse the direction of every line (or selected lines if any are selected) in the chosen feature class or layer.
This can be used from the toolbox in ArcMap, or using the syntax outlined in the ArcGIS Help site link above:
in_features | The input line feature class or layer. | Feature Layer
In arcpy this would look something like this: