Very basic Instantiate Question

moveDirection = transform.TransformDirection(Vector2.right); That should work but I'm not 100% sure because I only know javascript...

You want to create a float variable to keep track of how much time has passed since your last shot. In your update do something like this

 float timeElapsed = 0.0f;
 
 void Update(){
     timeElapsed += Time.deltaTime;
 } 

Then in your shooting code check if timeElapsed is >= your firingRate and reset timeElapsed to 0 every time you successfully shoot a bullet. For example setting firingRate to 0.25f would allow a shot every 250 milliseconds.

Edit1: Also, you can use transform.forward to get the current object's forward direction.

I'd personally recommend moving your shooting code to a separate script though. This makes debugging and modification easier.

Edit2: Another tip: you can make a new gameObject with a transform component and make it a child of your player. You can then put your shooting script(should be seperate form the movement script) on this new transfrom object and adjust it's transform.localPostion. This allows you to easily change the position of where your bullets come from. Just make sure that this child object's rotation is set so that it's forward direction is the same as the parent's forward direction.

I'm not sure about the cursor issue. Do you have any scripts that turn the cursor on or off and/or locks the cursor? Check your player settings for your build and see if there is anything that needs to be changed.