What is the impulse response for 2D functions
As pointed out by Jason R. in his comment, you can easily apply an impulse $\delta(m,n)$ (with $\delta(m,n)=1$ for $m=n=0$, and zero otherwise) to the systems and see what the output is. However, for those two examples the response to an impulse does not say much about the general behavior of the system.

There is an important class of systems called Linear Shift Invariant (LSI) systems which are completely characterized by their impulse responses. Linearity means that if $y_1(m,n)$ is the response to an input $x_1(m,n)$ and $y_2(m,n)$ is the response to input $x_2(m,n)$, then the response to the input $ax_1(m,n)+bx_2(m,n)$ is given by $ay_1(m,n)+by_2(m,n)$. If you check this condition you will find out that your first system is not linear.

Shift invariance means that if $y(m,n)$ is the response to $x(m,n)$ then $y(m-k,n-l)$ is the response to the input $x(m-k,n-l)$. Your second system does not satisfy this condition.

So none of the two systems is LSI and therefore neither of them can be characterized by their impulse response.