Linearity and shift-invariance of 2-D system on lattice
Well, actually you only need to insert the system $H$ into the definition of linearity and check, if it holds:

$$\begin{align}H\{af_1[x]+bf_2[x]\}&amp;=\sum_y\phi[y](af_1[x-y]+bf_2[x-y])\\
&amp;= a\sum_y\phi[y]f_1[x-y]+b\sum_y\phi[y]f_2[x-y]\\
&amp;=aH\{f_1[x]\}+bH\{f_2[x]\}\end{align}$$

So, the linearity holds. There is nothing special about the fact that the system is 2-dimensional or similar. You just follow the rules for the calculation, and you end up with the statement of linearity. 

A similar thing can be carried out for the shift-invariance.