Rotate an object until its x axis lies in a plane perpendicular to a given vector...

Well, first to answer your first question on how to project the object's local-x-axis onto the ground place defined by a ground normal. Unity now has the method Vector3.ProjectOnPlane which should do exactly what you asked for. It simply does this:

 public static Vector3 ProjectOnPlane(Vector3 vector, Vector3 planeNormal)
 {
     return vector - Vector3.Project(vector, planeNormal);
 }

So it actually projects your vector onto the normal and then simply subtracts that part from the original vector which will project it onto the place defined by the normal vector.

"Vector3 Project" on the other hand simply does this:

 public static Vector3 Project(Vector3 vector, Vector3 onNormal)
 {
     float num = Vector3.Dot(onNormal, onNormal);
     if (num < Mathf.Epsilon)
     {
         return Vector3.zero;
     }
     return onNormal * Vector3.Dot(vector, onNormal) / num;
 }

Since the normal vector is involved two times as a factor we don't have to normalize the normal vector but can simply divide by the square magnitude (num).

Finally another way is to simply use the Cross product between your normal vector and your forward vector. This gives you the "right" vector that you're after as well. Unity uses a left-handed-system so the left-hand-rule applies:

 Vector3 right = Vector3.Cross(normal, transform.forward).normalized;

Of course when you pull up over 90° right will flip to the other side as you officially are now upside down.

To get the relative rotation you might use Quaternion.FromToRotation. However i guess you want to apply a torque instead of rotating it manually, right?. So all you have to determine is:

• how far you have to rotate (the remaining angle) so you can adjust the rotation speed

• Which direction you have to rotate.

To get the remaining angle you can simply use Vector3.Angle with your object's right vector and the calculated target right vector.

To determine the direction you have to rotate, simply use the Dot product between the surface normal and your right vector. If the result is positive you have to rotate clockwise (as seen from behind looking along forward) and if it's negative you have to rotate counter-clockwise.

Note: you might disable the alignment if you almost go straight up or down (+-70°) to avoid constant realignment by 180° each time you cross the zenith.

     void $$anonymous$$ouse$$anonymous$$ove(float x_rot, float y_rot){
         //x and y values taken from mouse input;
         RaycastHit align; 
         var qRot = transform.rotation;
         qRot = Quaternion.AngleAxis (x_rot, transform.up) * qRot;
         qRot = Quaternion.AngleAxis (y_rot, transform.right) * qRot;
         if (Physics.Raycast (transform.position, transform.up * -1f, out align, 10f)) {
             //Check for surface beneath player, get right x axis projection on plane parallel to surface;
             Vector3 target = Vector3.ProjectOnPlane (transform.right, align.normal);
             //Check angle between right axis and projection; if greater than 1 degree deter$$anonymous$$e direction of rotation
             if (Vector3.Angle (target, transform.right) > 1f){
                 //construct reference plane, check if right axis is above or below plane 
                 refPlane.Set3Points (transform.position, target, transform.right);
                 if (refPlane.GetSide (Vector3.Cross (target, transform.right)))
                     z_rot = -1f;
                 else
                     z_rot = 1f;
             } else
                 z_rot = 0f; 
         } else
             z_rot = 0f; 
         qRot = Quaternion.AngleAxis (z_rot, transform.forward) * qRot; 
         transform.rotation = qRot;