Can a coordinate system overlap an ArcGIS Shapefile?
Even if your shapefile were in different coordinate system, ArcGIS should project them on the fly so that they overlap properly.
Why do layers with the same coordinate system not overlap?
My hypothesis is that your 30 m shift is due to datum difference but that your original shapefiles are correctly georeferenced (each in a different projection system). If selecting a transformation does not improve your accuracy, then I would add a third layer (e.g. a basemap) and try to find out which one is wrong.
How to project a shapefile in ArcGIS Desktop?
Essentially, your all the features of your shapefile are now fitting in a distance of 360 feet horizontally and 90 feet vertically. The proper procedure, to be performed on the original shapefile in the geographic coordinate system, is to use the Project Tool. For the Projection, choose State Plane, NAD 1983 California Zone 6 Feet.
Can you overlay ArcGIS on any other layers?
It should also overlay correctly on any other layers, provided they have a projection defined and that your Data Frame in ArcMap has a projection defined. Although you have set the projection of your 3 other layers by the properties menu in ArcCatalog, you have failed to re-project the data to the proper coordinate system.
Can you select a different coordinate system in ArcGIS?
Selecting a different coordinate system that way does not change the coordinate system. You did the same thing the Define Projection tool does, which is to basically tell ArcGIS “This layer is in the specified coordinate system”.
How to export a coordinate system in ArcGIS?
On Coordinate System of Input Coordinates field, select current CRS from data. Here the CRS is defined. One can export the layer to a shapefile by right clicking in the layer and then, Data –> Export Data. Or continue (from item 3.2) to reproject: