remove the word "copy" at duplicate image

Using Adobe Bridge (included with PS install) you can use the Batch Rename feature to group rename. I do this often when I have lots of copies in a directory. I highlight all of those files with the word copy, and then set it to a new naming sequence of my choice.

As c.pfaffenbichler wrote for layers, this is a native option… However for duplicating a document, a script like this would be required:

#target photoshop
var doc = app.activeDocument
doc.duplicate(doc.name.replace(/\.[^\.]+$/, ''), false);

Prepression: Downloading and Installing Adobe Scripts

You can assign a keyboard shortcut to the script and optionally disable/hide the native menu command for Image > Duplicate so that it is not used.

c.pfaffenbichler is right on the money with this one.

With a Script.

Or are you talking about Layers?

Edit: For Layers is can be turned off in the Layers Panel Options’ »Add "copy" …« .