How may connected components does the complement of the conic surface (extending on both sides) have in the three dimensional space?
I think the connected components is two, but am not convinced about it. Like, I sometimes think it is one whole block that is somehow connected by circular shape of the nappes, but again it is just my presumption. So, what is the right number of connected components.
The question is unclear because planes are $2$-dimensional. I'll assume you meant $3$-dimensional space and your cone looks like the surface given by $x^2+y^2=z^2$. In this case, the complement has $3$ components:
the part inside the half of the cone above the vertex, i.e., $x^2+y^2<z^2$ and $z>0$,
the part inside the half of the cone below the vertex, i.e., $x^2+y^2<z^2$ and $z<0$, and
the part outside the cone, i.e., $x^2+y^2>z^2$.