The mouse position is in theory external to the game world itself. Therefore, the mouse position relates simply to X and Y co-ordinates of the screen space in which you are interacting with your game (IE width:height of your game).
What you're asking seems to be more of "How do I model the mouse position in my game world?"
As noted by Mario, Camera.main.ScreenToWorldPoint(Input.mousePosition) will convert your mouse position to world co-ordinates. However, this assumes that your main camera is where you want to convert to. In reality, you want to call the ScreenToWorldPoint method on the Camera that is rendering whatever space it is you are wanting to interact with. For example, you may have your main game world at (0, 0, 0) but you may be rendering your GUI on top using a separate camera that renders objects at (-5000, 0, 0).
To answer your question, to model the mouse z position it should simply be the same z value as your Camera. You can then perform calculations on that value to suit your particular needs.
IE:
1) mouse.position.z = mouse.position.y - These are entirely different. Now you're just using an arbitrary value
2) Distance between camera and object - That's a calculation made from your original object.position.z and original mouse.position.z. Not your actual z value.
3) See 2.
4) See 1.