The net gravitational force is zero: https://en.wikipedia.org/wiki/Shell_theorem
If the box were a perfect sphere then yes it would have zero net gravitational pull on the black hole, but the black hole would still have an extremely strong pull on it. And that's still assuming it's perfectly centered which would be almost impossible to maintain.
No, it would not. Newton's third law: if an object has zero net pull on the black hole, the black hole also has to have a zero net pull on the object. The shell theorem is more general than you are assuming: it doesn’t matter if the heavier object is inside or outside, the surrounding object doesn't need to be spherical (it can be any shape as long as it fully envelopes the black hole) and the black hole doesn't need to be centered. I know it is a counter-intuitive result, but it can be easily proven with calculus.
See for instance this sentence about Dyson shells: "Such a shell would have no net gravitational interaction with its englobed star (see shell theorem)" https://en.wikipedia.org/wiki/Dyson_sphere#Dyson_shell
Edit: Related, what is bigger? The gravitational force the Moon exerts on Earth or the gravitational force the Earth exerts on the Moon? Veritasium has a nice video explaining the answer: https://www.youtube.com/watch?v=8bTdMmNZm2M
Cage's material will still get attracted to the black hole and close to the event horizon structural forces exerted by gravity will be hard / impossible to counteract with current materials: https://en.wikipedia.org/wiki/Strength_of_materials
Reading the exact passage you quoted, "the compressive strength of the material forming the sphere would have to be immense to prevent implosion due to the star's gravity."