Sphere of circumferential radius $r$

Question

What do you then mean by constructing a sphere of circumferential radius $r$ centered on the black hole? Is that sphere a 2D surface?

Answer 1

This is as discussed in my answer to your previous question: Angular coordinates are unaffected?

The phrase circumferential radius means the radius you get by dividing the circumference of the sphere by $2\pi$ i.e. it is the Schwarzschild $r$ coordinate of the sphere.

The sphere is indeed a two dimensional manifold. That is the $r$ coordinate is fixed and the $t$ coordinate is ignored so points lying in the sphere can be labelled using just the $\theta$ and $\phi$ coordinates.