I believe I figured out the reason for delete and rename (for feature classes) being grayed out, which was that I had an editing session open and had forgotten to stop the editing session.
However, now I am finding that I cannot add feature classes or rasters to the geodatabase using Spatial Analyst tools (Extract by mask or Contours). I can add rasters or shape files to the folder located on the network drive but not to the gdb within that folder; I do get error messages related to permissions. It appears to be an issue limited to Spatial Analyst tools creating features or rasters onto the gdb located on a network folder. If I use Spatial Analyst Contour tools to create contours from a raster DEM with the destination into the gdb in the network folder, there are no error messages but it creates an empty polygon feature class with a table type symbol in the gdb; but I can create a contour feature class with the Spatial Analyst tools onto a gdb on my computer hard drive.
I created the geodatabase on the network folder by simply right clicking on the folder to create a new gdb. I am not using an enterprise version so I don't have a Version Manager view. I can use the Spatial Analyst tools to create feature classes within gdb's on folders that are on my computer hard drive; but I cannot copy those feature classes onto the gdb located on the network folder. I read something about the need to load MS SQL Express but that is getting more complicated than I had hoped it would need to be. I didn't realize that working from an NAS should cause such complications. I am not sure of a work around to work on the hard drive but save it onto the network drive, which gets backed up for crash security reasons.