What do I with the sampling period when I inverse $\mathcal Z$-transform?
There should probably be a $z^{-2}$ term with the $0.3$. It looks like a typo. 

Once you've transformed to the $z$ domain (or discrete-time), and are not interested in transforming back to the continuous-time domain, there is no real need to carry the $T_s$. Just refer to $e[k], y[k]$, etc.