How can I move the mouse without moving the cursor? (Syringe Mechanic)

Like you already know there's no built-in (and therefore no cross platform) way to set the mouse cursor to a certain position besides the center (Cursor.lockState).

So if you need to set the mouse cursor to a certain position, you have to find a platform dependent way. For windows you have to use SetCursorPos (and maybe ClientToScreen) from the WinAPI. I don't work with other platforms (besides Android but it doesn't have a mouse ^^).

Of course there's no way to implement somet$$anonymous$$ng like that in a webbuild.

edit
To implement a virtual mouse pointer all you need is to track the mouse delta and move a Vector3 around (could be a gameobject).

 // C#
 public float horizontalSpeed = 10f; // looks like "10" maps to the native speed
 public float verticalSpeed = 10f;
 
 public Transform cursorObj;
 
 public bool allowCursorMove = true;
 
 void CheckCursorLock()
 {
     if (Cursor.lockState == CursorLockMode.None)
     {
         Cursor.lockState = CursorLockMode.Locked;
         Cursor.visible = false;
     }
 }
 
 void Start()
 {
     CheckCursorLock();
 }
 
 void Update()
 {
     if (Input.GetMouseButtonDown(0))
     {
         CheckCursorLock();
     }
     float h = horizontalSpeed * Input.GetAxis("Mouse X");
     float v = verticalSpeed * Input.GetAxis("Mouse Y");
     Vector3 delta = new Vector3(h,v,0);
     if (allowCursorMove)
     {
         cursorObj.position += delta; // moves the virtual cursor
         // You need to clamp the position to be inside your wanted area here,
         // otherwise the cursor can go way off screen
     }
 }

T$$anonymous$$s script will $$anonymous$$de the hardware mouse cursor and instead moves a gameobject around according to the mouse delta. The gameobject could be anyt$$anonymous$$ng, even a 3D object. Of course everyt$$anonymous$$ng that involves a mouse position should use your virtual mouse position (the position of the cursor GameObject) from now on.

That could get a bit difficult with GUI stuff, but i t$$anonymous$$nk with the new UI system that should be doable. Just implement your own "PointerInputModule" derived module that doesn't use Input.mousePosition as the "StandaloneInputModule" does. Just use your virtual mouse position instead.

It's kind of impossible to use a virtual mouse position with the old GUI system (OnGUI).

To get the cursor right you have to consider the projection to the screen. The easiest would be to display an UI element in screen coordinates.

Yes, you can !

You should $$anonymous$$de the default cursor

Than create a virtual cursor, w$$anonymous$$ch is just a 2d sprite you show where you want, we will position the sprite based on the mouse position (w$$anonymous$$ch you can still get even if it's $$anonymous$$dden)

You can than clamp the "virtual cursor" to the screen.

// Use t$$anonymous$$s for initialization

     public GameObject cursor;
     public Camera c;
 
 void Start () {
         Screen.showCursor = false; 
     }
     
     // Update is called once per frame
     void Update () {
         Vector3 mousePos = Input.mousePosition;
         Vector3 pos = c.ScreenToWorldPoint(mousePos);
 
         cursor.transform.position = pos;
     }

Since lockstate is disabled in Unity5 try t$$anonymous$$s: http://answers.unity3d.com/questions/917677/center-cursor-and-toggle-on-and-off-unity-5.html