Can Ampere's law be applied to 3D loop Usually in textbooks Ampere's law is just illustrated using 2D loops which forms a plane. Can the law be applied to 3D loops which cannot form a plane surface within the loop?
In a 3D loop which surface should we count when we count the current? In 2D case it is easy because there can only be a plane and the current is just that cut through the plane but if the loop is 3D that may be more then one possible surface formed by the loop, which section of the current should be counted? 
 A: What do you consider to be Ampere's law?
I think the RHS is not the current that cuts some plane, it is actually a surface integral of the current density, over any surface that is bounded by the loop.
In other words it does not matter what surface you define ; so long as it is bounded by the loop, Ampere's law works just fine. So usually, you choose the surface which makes the RHS easiest to evaluate.
EDIT:
The reason that it works in this way can be ascribed to the proof of Stokes's theorem. The differential form of Ampere's law is
$$ \nabla \times \vec{B} = \mu_0 \vec{J},$$
where $\vec{B}$ is the magnetic field and $\vec{J}$ is the current density (and I am deliberately ignoring the displacement current - i.e. assuming a static problem).
Stokes's theorem tells us that
$$ \oint \vec{B} \cdot d\vec{l} = \int (\nabla \times \vec{B}) \cdot d\vec{A}, $$
where the left hand side is a closed line integral and the right hand side is the surface integral over any surface bounded by the closed loop.
The combination of these two leads to the integral form of Ampere's law
$$ \oint \vec{B} \cdot d \vec{l} = \mu_0 \int \vec{J} \cdot d\vec{A}$$
Thus the proof you seem to require is in fact the proof of Stokes's theorem, which is a well-established theorem of vector calculus. In summary, the boldface property arises because you can split any surface into small square elements; the line integrals around these squares cancel on their adjoining sides except where they reach the edge of the surface bounded by a closed loop. Thus the net result is a line integral around the close loop, regardless of what surface area it bounds. This result will be discussed in any book that needs to deal with vector calculus. 
A: In the case of a planar loop, the surface that it bounds does not have to be a plane.  It could bubble out.  Ampere's Law is still valid (assuming the other conditions for validity are met).
Ampere's Law applies to any loop, and any surface bounded by the loop.  I guess I'd better add that I don't know what happens in pathological cases where, for example, the loop crosses itself.  
