Is it LTI system? given $H(z)$ is LTI
Lets say that $x_1[n]=\delta[n]$ and $x_2[n] = \delta[n-1]$, where $\delta[n]$ is Kronecker delta and that $y_1[n],y_2[n]$ are the output signals for $x_1[n],x_2[n]$.
The downsampling operator discards every other sample. Without the loss of generality, lets assume that we are keeping the even samples and throwing the odd ones. Therefore, after downsampling $x_2[n]$, it is all zeros, which means that $y_2[n] = 0$ for each $n$. Now since you say that $h[n]$ is not all zeros, it is guarantied that $y_1[n]$ will have some non-zeros values.
In other words, $y_1[n]\ne y_2[n]$ for some $n$, and you system is not TI.