Rotating a 3D character on when they move - unity3d

Question

I want to start with a little 3D platformer on Unity. When I move, I want the character looking to the moving direction. So when I press Left/"A" I want the character instantly turning left and walking forward. Same for the other directions. The problem is that the character turns back to the default rotation when I leave the key.
The important code:
private void FixedUpdate()
{
float inputX = Input.GetAxis("Horizontal"); // Input
float inputZ = Input.GetAxis("Vertical"); // Input
if (GroundCheck()) // On the ground?
{
verticalVelocity = -gravity * Time.deltaTime; // velocity on y-axis
if (Input.GetButtonDown("Jump")) // Jump Key pressed
{
verticalVelocity = jumpPower; // jump in the air
}
}
else // Player is not grounded
{
verticalVelocity -= gravity * Time.deltaTime; // Get back to the ground
}
Vector3 movement = new Vector3(inputX, verticalVelocity, inputZ); // the movement vector
if (movement.magnitude != 0) // Input given?
{
transform.rotation = Quaternion.LookRotation(new Vector3(movement.x, 0, movement.z)); // Rotate the Player to the moving direction
}
rigid.velocity = movement * movementSpeed; // Move the character
}
The second thing is, at
transform.rotation = Quaternion.LookRotation(new Vector3(movement.x, 0, movement.z));
there is 0 on the y-axis. It says the viewing Vector is zero. When I pass in another number like movement.y the character tilts to the floor. So I do not know what to pass in there.

Answer 1

As for resetting when you let go of the key: your line
if (movement.magnitude != 0) // Input given?
is a good idea, but there's a good chance your controller is reporting slightly off of 0, so your character's direction will change even when you're not actually moving. I would change this to
if (movement.magnitude >.1f) // Input given?
or some other number close to (but not exactly) zero. While working on this, I would add Debug.Log(movement.magnitude); to this function and make sure that the values are in the range you expect.
On the second topic:
while it is important to have verticalVelocity in your movement vector for when you apply it to rigidBody.velocity, you don't want it in your character facing vector. If you want your character to only look in a single plane, it makes perfect sense to only give it two dimensions to consider; adding a third dimension would make it look at the sky or the ground as you mentioned. Furthermore, I would change your input-checking line to use this as well, because you only want to change facing based on whether the character is moving horizontally or not. This would make your code look something like this:
Vector3 movement = new Vector3(inputX, verticalVelocity, inputZ); // the movement vector
Vector3 horizontalMovement = new Vector3(inputX, 0f, inputZ);
if (horizontalMovement.magnitude != 0) // Input given?
{
transform.rotation = Quaternion.LookRotation(horizontalMovement); // Rotate the Player to the moving direction
}
rigid.velocity = movement * movementSpeed; // Move the character
And one final note, when you are grounded, you might want to set verticalVelocity to 0, rather than -gravity*deltaTime. This error may not be visible (the physics engine will push your character back up out of the floor), but if the user alt-tabs and there's too long between frames, your character will teleport through the floor!
Good luck.