This how far I've gotten. I hope someone with more a recent education than I can take correct what I have wrong and finish it up.
Here are pieces I think will help:
Name the 4 points A, B, C, D and their coordinates as $(x_A,y_A,z_A)$, etc.
Name the ship S and its coordinates are $(x_S,y_S,z_S)$
Name the planet P and its coordinates are $(x_P,y_P,z_P)$ with a radius $r_P$
For convenience, set the center of the planet $P$ to be at the origin.
In addition, orient the axes such that the ship $S$ is directly above the planet along the z-axis.
This tells us that $x_P=y_P=z_P=x_S=y_S=0$
The general equation for the distance between two points in 3D space is
Therefore, the distance between the ship $S$ and point $A$ is
Now, since the planet's center is at the origin, we know $x_A^2+y_A^2+z_A^2=r_P^2$
We can plug that in and square both sides to get
It is trivial to expand that to the other 3 points
Right now, we're sitting at 4 equations and 7 unknowns ($r_P,z_S,z_A,z_B,z_C,z_D,D_D$). That's close but we're not there yet. We can play around with the formulas for $D_A$ and $D_B$ because they're values are known.
$$D_A^2+32 = 49+32=81=D_B^2$$
We can do the same thing for $D_A$ and $D_C$ as well as $D_B$ and $D_C$
If we solve all three for $z_S$, we can set them equal.
This gives us the relationship
Let's look at the second observation. The formula for the surface area a spherical cap $A_P$ that is distance $h$ from the origin and the area of a circle $A_C$ are given by
We're told those are equal so we can drop the $\pi$ to get the equality
Unfortunately, this introduces only 1 new equation and 2 new unknowns. I think the solution will involve some trig to find $h$ based on $r_P$, $z_S$, and tangent lines but I don't have the brainpower for that right now.
I'm sure that the probes landing on opposite sides matters but I couldn't get it to do anything useful because it added more unknowns instead of removing them. Here's what I wrote for that section:
We also know that the probes landed on opposite sides of the sphere. That means the line connecting them passes through the center which means it is of length $2r_P$. That gives us the following for the distance between probe $A$ and $B$. (Here I have to assume that the first two distances are for the first two probes. If that's not the case, this section must be reworked.)
(I'm not super-confident in how the $\pm$ comes in once you square both sides but I think it has to be considered somewhere.)
We can do the same thing for the other two probes