Difference equation of second order system with zero

Question

I saw from lecture notes that difference equation of a first order system is like this:

(1) $$ G(s)=\frac{K}{\tau s+1} $$ (2) $$ G(z)=\frac{K(1-A)}{z-a}, A=\exp (-T / \tau) $$ (3) $$ G(z)=\frac{K(1-A) z^{-1}}{1-A z^{-1}}=\frac{Y(z)}{X(z)} $$ (4) $$ y(k)=A y(k-1)+K(1-A) x(k-1) $$

1. What happens between (3) and (4) ? It looks like inverse Z-transform but according to table $$ \frac{z^{-1}}{1-a z^{-1}} $$ should transform into this: $$ a^{k}, k=1,2,3, \ldots $$
2. How can I find the difference equation of systems like this: $$ T(s)=\frac{A s+B}{s^{2}+C s+D} $$