What is the average temperature of the surface of this planet?

There is a very basic solution that takes advantages of some of the symmetries you can impose on the problem.

$T\left(x,y,z\right)=x^2+2y^2+z^2+2xy-2yz$

Now we can simplify this to

$T\left(x,y,z\right)=400+y^2+2xy-2yz$

Since $x^2+y^2+z^2=400$

But we still have a lot of left over terms to get rid of and we can do this by taking advantage of symmetry.

By switching the order of our inputs we can get:

$T\left(x,y,z\right)=400+y^2+2xy-2yz$

$T\left(z,x,y\right)=400+x^2+2zx-2xy$

$T\left(y,z,x\right)=400+z^2+2yz-2zx$

So we have

$T\left(x,y,z\right)+T\left(z,x,y\right)+T\left(y,z,x\right)=1600$

For any $x,y,z$

This is helpful because if $(x,y,z)$ is on the sphere then we know both $(z,x,y)$ and $(y,z,x)$ are also on the sphere.

Thus, for all points on our sphere such that $x\neq y\neq z$ we can group these points into groups of $3$ such that the sum of their temperatures is $1600$. This means the average temperature on the surface of the planet is $1600/3$ degrees.

Note that we do not have to take into account the cases where $x=z$ or $y=z$ or $x=y$ since they make up such a minuscule part of the actual sphere.