Determining a system's causality using its impulse response

I have the following input-output relation for a system:

$$y(t) = Odd Part Of [x(t)]$$

My question is: Is the system causal?

What my approach has been:

I expressed $y(t)$ alternatively as:
$$y(t) = frac{x(t) – x(-t)}{2}tag{1}$$

Here, when I substitute $x(t)$ with the impulse function, I get the impulse response as $0$ because the impulse function is an even function. Its odd part is $0$. This leads me to believe that the system is causal as the impulse response is zero for negative time.

However, when I substitute $t$ with $-t$ in Eq. $(1)$, I find that for negative time, the output depends on the input at a future time. This would lead me to believe that the system is non-causal.

So my question is actually two-fold here:

1. How do I reconcile the two seemingly contradictory results?

2. Why is the impulse response of the system 0? What’s the difference between this system and having no system at all?

Any help/pointers would be sincerely appreciated.