Am I right that if I want to choose layers for QGIS processing tools I always have to choose the shapefiles via the path directory, and that I can't drag and drop them from the layer window on the left like in ArcGIS Desktop?
I haven't seen the drag and drop functionality you mention, I agree that would be nice.
So I suppose to answer your question "can you drag and drop layers like in ArcGIS?" is no, not at the moment.
But at the same time you don't always have to specify the full path to a shapefile via the path directory like you suggested above.
You can still speed up your data entry by using the "Merge Vector Layers" from the processing toolbox shown below.
It does the same task as the tool you mentioned in your comment, however it gives you the nice option of being able to add the layers which are already in your map document. When you click on "Layers to merge" you can toggle on all the layers, or select them individually, check out the dialog below.