How to check if h(n) is causal, stable?
Firstly, you do not need to specify an input $x(n)$ to know whether the system is stable or causal, these are properties of the system itslef, I will interpret $h(n)$ you mention as a array describing the system response.

For causality in time domain perspectivefor an LTI system check if $h(n) =0 $ for all $n &lt; 0$, that is the $h(n)$ is essentially a right sided sequence.

For causality in Z domain perspective equivalently, the Region of Convergence is the right side of the outermost pole (outermost in terms of the highest magnitude pole in z domain)

For stability in time domain perspective, the system response $h(n)$ should be absolutely summable, since you have only finite coefficients, this system is stable, any FIR filter is always stable.

For stability from Z domain perspective, the region of Convergence must contain the unit circle.

For stable and causal system all these conditions hold at the same time.