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. But:
Will two surfaces having the same boundary have the same surface integral?
Yes.
Any two surfaces that have the same boundary will have equal amounts of flux. For example, if you have a magnet inside a large sphere and slice off a section off the top of the sphere so that you have a very small hemisphere (so that it almost looks like a flat disc) and a larger one being the rest of the sphere, that both of these spheres have the same boundary will mean they have equal amounts of flux.
The number of lines of forces going through the larger hemisphere will be significantly higher than the smaller section, but the number of lines of force coming back into the larger section will cancel most of those leaving so that the net flux going through both surfaces turns out to be equal.