A line can be curved.

A plane can be curved.

A 3D space can be curved.

I am not sure, but a N dimensions space can be curved: I think the property of being curved is valid for every N dimension space, where N ≥1.

Being curved doesn’t need another dimension. The surface of a sphere doesn’t need the 3rd dimension to curve inside. Not everything curved, closes.

Flat, semantically, can be used in more than one context, surely it is not used just for 2D planes. Those who say that earth is flat, don’t imply that earth is a 2D object. In the context of the article flat means not curved. I don’t know the exact definition of flat for a 3D space, but a 2D space is flat if the sum of the angles of a triangle is 180 degrees.

Those are mathematical concepts, not physics, and thus not questionable (although it is definitely questionable my geometry knowledge).

After general relativity, most scientists believe that mathematical curved spaces are good models in order to explain some phenomena, among them gravity.

You can believe what you prefer regarding physics, but please stop questioning mathematics.