We can estimate the pressure pretty easily, since both water ice and $CO_2$ ice are flexible enough that the pressure and weight must be more or less in balance after a while. At the top of the $CO_2$ layer, we have a 20m column of ice above us, so the pressure is $20 \rho g$ where $\rho$ is the density of the ice ($10^3 kg/m^3$ near enough) and $g$ is local gravity ($3.7 m/s$ according to wikipedia).
So we get about $75 kPa$, roughly 0.75 Earth atmospheres. The density of dry ice depends on conditions, but is around $1500 kg/m^3$, so at depth $d\, m$ inside the dry ice layer we would find a pressure of $75000 + 1500\times 3.7 \times d\,Pa$ which is about $75 + 5.6d\, kPa$. So at the bottom of that dry ice layer we get about $1.8 MPa$ (18 Earth atmospheres).
I have no idea how to estimate the temperature.