So the problem I have is that I have a sphere $$x^2+y^2+z^2=1$$ and a plane $$x+y+z=0$$ that goes through the center of the sphere and creates a circle. Now, what I tried was plugging in $$z=-x-y$$ into the equation for the sphere and got $$x^2+y^2+yx= 1/2$$ But this isn't a circle, it's an ellipse so somewhere I must have gone wrong.
Also, the answer is supposed to be in spherical coordinates. Should I just convert the final answer to spherical coordinates then? That would mean I would get $$ r^2\sin(\phi) + r^2 \sin(\phi)^2\sin(\theta)\cos(\theta) = 1/2$$ after you plug in $$\sin(\theta)^2 + \cos(\theta)^2 = 1$$. Is that a correct transformation into spherical coordinates?
The intersection of the sphere $x^2+y^2+z^2=1$ with the plane $x+y+z=0$ is simply described by those two equations. It certainly isn't $x^2+y^2+xy=\frac12$, since that defines a surface, not a line. You made no mistake in your computations, but you only have$$x^2+y^2+z^2=1\wedge x+y+z=0\implies x^2+y^2+xy=\frac12,$$and this implication is not an equivalence.
And the equation in spherical coordinates is$$\left\{\begin{array}{l}\rho=1\\\cos(\theta)\sin(\phi)+\sin(\theta)\sin(\phi)+\cos(\phi)=0.\end{array}\right.$$