z-transform causality properties: negative coefficents are zero ($x[-1]z^1=0$, $x[-2]z^2=0$, ...)
No. The system response is usually $H(z)$. If $H$ is causal, then it has no non-causal coefficients.  $X$ may very well be a signal that exists for all time (and so existed before time 0) and, therefore, have non-causal coefficients.  That means $Y$ will also have non-causal coefficients.

If $Y$ is taken as the system and $Y$ is causal, then your statement is true. But usually $Y$ is the output, $X$ is the input, and $H$ is the system response.