Finding Z-parameters from a two-port network and using them to find S-parameters
calculate the current equations when ignoring the sides on each scenario (2Ω and 10Ω),


I'm not sure what you mean about ignoring "the sides". You shouldn't ignore any component in your analysis. Of course, if you find there is no current flowing through a resistor, then the voltage across it will be zero.


  but I'm not exactly sure what I should get from that or how I should proceed on with that.


The Z parameters allow you to calculate

$$\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}Z_{11} &amp; Z_{12} \\ Z_{21} &amp; Z_{22}\end{bmatrix}\begin{bmatrix}i_1\\i_2\end{bmatrix}$$

meaning if you know the two port currents, you can calculate the port voltages.

So to work out the Z matrix from circuit analysis, you apply a fixed current at each port (one at a time), and see what voltages result.

If you apply 1 A at port 1, and measure a certain voltage \$v_2\$ at port 2, then that immediately gives you the value of \$Z_{21}\$, for example.