In the equation $$\Phi=\int_{\mathbf{s}}\bf B\cdot dA$$ there is a dot product between the magnetic field and the normal area vector. This is evaluated at all points on the surface $\mathbf{s}$ and each one of these dot products depends on the angle between $\bf B$ and $\bf dA$.
For surfaces that are of different shape or orientation, the dot product will be different at points on each surface. So even if the total surface area between two surfaces is the same, the value of $\Phi$ will differ in general.
So a paraboloid and the section of a sphere will not have the same value for $\Phi$ even if the total surface areas and $\bf B$ (which includes the direction of $\bf B$ of course) are the same for each.
The same logic applies to the rest of your question.