I want to create a watershed shapefile with areas, streams and pour points in it.
Is there a way to merge the areas (polygons) with streams (polylines) and pourpoints (points)?
Nope. GIS datasets can only support one type of geometry at a time. What you can do however is package those layers into a single Geopackage (or spatialite, or geodatabase). This is a lightweight database which can store multiple layers within a single file. GIS will treat them as seperate layers so they won't be merged, but it's tidier for storage and distribution.
With shapefiles that is not possible. You can have all kind of geometries in the same table in PostGIS, GeoPackage, SpatiaLite and Oracle if the table is created without geometry constraints. The native format (JML) of OpenJUMP can also contain any kind of geometries as a mixture. Another question is which softwate can handle such mixed data. QGIS and ESRI can't while OpenJUMP can.
If you were not tied to use ArcGIS you could open your point, line, and polygon shapefiles with OpenJUMP, select all three and use the command "Combine Layers" and that's all.