Is there a way to build an object through intersection? I am really stumped by this logo I am trying to animate, and I am not sure what to search for or how to phrase a search query. I have been trying to use the Boolean Modifier with this approach, but the problem is that the object I am trying to animate is gone after the mesh I am intersecting the object with moves away from the object.
The setup that I have got so far:

Start with a torus, and then add a sphere that is slightly big as
the thickness of the torus.
Create a circular path such that the path roughly goes through the
center of a circle made from a cross-section of the torus.
(Basically, the path is inside and centered around the torus).
Constrain the sphere to follow the circular path. Now I have a
sphere that goes around a torus.


Now I want to do this:

As the circular sphere goes around the torus, the intersection of the
sphere and the torus is revealed.


As the sphere moves away from the newly revealed part, it stays there
and does not disappear.

A nice bonus is if there is a way to add a glow to the object only when it
is inside the sphere.
You almost have it.  In your last photo if you select the sphere and press Ctrl + A and apply its position, it'll move the object origin back to the center.


Then you can just rotate it around the Z axis.

Whatever material properties you set for the torus will only be shown for the visible portion, including volumes.