Is it possible to change Layer 0 to the old Background permanently in Photoshop CC?

Right. So long as the OP understands that "Background" (an actual flat background layer) is not the same as "Background" (a renamed floating layer).

You can use the name Background on a transparent layer (or layers), so a simple action step on the bottom layer should work.

Thee are many possible solutions for your problem. How you are getting Layer 0 in your document? By converting Background layer to normal? Record action for that job and to rename it to Background autmaticaly or record action to rename bottom most layer to Background and actions should work. By the way, you can set up Script Events Manager to automatically reame bottom most layer to Background or to convert it to Background for what you need to record action which will play on event Open Document. 

There are and other solutions to change / modify actions but that will require more time and effort.

Why do you not use a Background Layer in your files, by the way? 

backgroundLayer to normal:

 try {
        var d = new ActionDescriptor();
        var r = new ActionReference();
        r.putProperty(stringIDToTypeID("layer"), stringIDToTypeID("background"));
        d.putReference(stringIDToTypeID("null"), r);
        var d1 = new ActionDescriptor();
        d.putObject(stringIDToTypeID("to"), stringIDToTypeID("layer"), d1);   
        executeAction(stringIDToTypeID("set"), d, DialogModes.NO);
        }
    catch (e) {   alert(e);  }

normal to background:

 var d = new ActionDescriptor();
        var r = new ActionReference();
        r.putClass(stringIDToTypeID("backgroundLayer"));
        d.putReference(stringIDToTypeID("null"), r);
        var r1 = new ActionReference();
        r1.putEnumerated(stringIDToTypeID("layer"), stringIDToTypeID("ordinal"), stringIDToTypeID("targetEnum"));
        d.putReference(stringIDToTypeID("using"), r1);
        executeAction(stringIDToTypeID("make"), d, DialogModes.NO);

No the name can't be changes, and it's not just a name, the background layer is very different than a normal layer and it will make actions and scripts work differently depending if it exists or not. Most actions use the layer's name as a reference, so you might be able to edit the action by double clicking on the step you want to change. You can then re-record that step.

You can change a Layer to a Background Layer (unless it already exists) via Layer > New > Background From Layer.