Rotation Question

The problem with the script you've posted is that you'll get movement inconsistency at higher speeds, also if you'd like to be able to rotate several objects individually in one scene it won't let you. It's not very clear what you want the outcome to be, therefore I've made two versions of the script:

Parented version

 private var smoothSpeed : int = 10; //How fast should the object return to position
 private var speed : int = 20; //Sensitivity for mouse movement
 private var mouseMovement : Vector2; //Stored mouse axis
 private var mouseToRot : Quaternion; //Rotation variable
 private var rotClamp : int = 90; //Clamped degrees
 private var selected : boolean = false; //Checks for selection
 
 function Start () {
     transform.parent = Camera.main.transform;
 }
 
 function Update () {
     if(Input.GetMouseButton(0) && selected) {        
         mouseMovement -= Vector2(-Input.GetAxis("Mouse Y")*speed, Input.GetAxis("Mouse X")*speed); //Set mouseMovement to mouse movement on left mousebutton down
     } else {
         mouseMovement[0] = Mathf.SmoothStep(mouseMovement[0], 0, Time.deltaTime*smoothSpeed); //Return to position
         mouseMovement[1] = Mathf.SmoothStep(mouseMovement[1], 0, Time.deltaTime*smoothSpeed); //Return to position
         selected = false; //Set selected to false as we no longer push the left mousebutton
     }
     mouseToRot = Quaternion.Euler(Mathf.Clamp(mouseMovement[0], -rotClamp, rotClamp), Mathf.Clamp(mouseMovement[1], -rotClamp, rotClamp), 0); //Use Mathf.Clamp to clamp degrees
     transform.localRotation=mouseToRot; //Apply rotation to Transform
 }
 function OnMouseDown () {
     selected = true; //Selected will go true on left mousebutton down on objects with a collider
 }

Copied rotation version

 private var smoothSpeed : int = 10; //How fast should the object return to position
 private var speed : int = 20; //Sensitivity for mouse movement
 private var mouseMovement : Vector2; //Stored mouse axis
 private var cameraRotation : Quaternion; //Stored camera rotation
 private var mouseToRot : Quaternion; //Rotation variable
 private var rotClamp : int = 90; //Clamped degrees
 private var selected : boolean = false; //Checks for selection
 
 function Update () {
     if(Input.GetMouseButton(0) && selected) {        
         mouseMovement -= Vector2(-Input.GetAxis("Mouse Y")*speed, Input.GetAxis("Mouse X")*speed); //Set mouseMovement to mouse movement on left mousebutton down
     } else {
         mouseMovement[0] = Mathf.SmoothStep(mouseMovement[0], 0, Time.deltaTime*smoothSpeed); //Return to position
         mouseMovement[1] = Mathf.SmoothStep(mouseMovement[1], 0, Time.deltaTime*smoothSpeed); //Return to position
         selected = false; //Set selected to false as we no longer push the left mousebutton
     }
     cameraRotation = Camera.main.transform.rotation; //Store rotation of Main Camera
     mouseToRot = Quaternion.Euler(-cameraRotation.eulerAngles.x+Mathf.Clamp(mouseMovement[0], -rotClamp, rotClamp), -cameraRotation.eulerAngles.y+Mathf.Clamp(mouseMovement[1], -rotClamp, rotClamp), cameraRotation.eulerAngles.z); //Use Mathf.Clamp to clamp degrees
     transform.rotation=mouseToRot; //Apply rotation to Transform
 }
 function OnMouseDown () {
     selected = true; //Selected will go true on left mousebutton down on objects with a collider
 }

The Parented version will make the object child itself and use the localRotation derived from the camera rotation. The Copied rotation version will store the cameras rotation values and apply to the final rotation of the object.

"I would like the area of the 3D object facing the camera to be the area that rotates upward and downward at any given time."

To add a rotation just create a floating variable that you add to the final eulers of the object:

 var extraRotation : float = 45.0;
 mouseToRot = Quaternion.Euler(-cameraRotation.eulerAngles.x+extraRotation+Mathf.Clamp(mouseMovement[0], -rotClamp, rotClamp), -cameraRotation.eulerAngles.y+Mathf.Clamp(mouseMovement[1], -rotClamp, rotClamp), cameraRotation.eulerAngles.z);

How you choose to control that variable is up to the implementation.

Hopefully it can get you started on something that is closer to what you want to achieve.