Would Faraday’s law still hold? According to what I understand about Faraday’s law, this is how it works:
$$\mathcal E_C = \oint_C \vec{E}\cdot d\vec{\ell} = -\frac{d}{dt}\iint_S \vec{B} \cdot d\vec{A}$$


*

*$C$ is an arbitrary closed curve

*$S$ is the surface bounded by $C$

*Each $\vec E$ is the electric field vector at each point around $C$

*Each $\vec B$ is the magnetic field vector that pierces each point in $S$

*Each vector $d\vec\ell$ has a magnitude $d\ell$, an infinitesimally small portion of the length of $C$, and a direction tangential to $C$ at each point on $C$

*Each vector $d\vec{A}$ has a magnitude $dA$, an infinitesimally small portion of the area of $S$, and a direction normal to $S$ at each point in $S$ (oftentimes selected to point to the same side as does the net magnetic field)

*$\mathcal E_C$ is the induced emf distributed around $C$

*$t$ is time


However, how would Faraday’s law hold—if at all—if it were extended to a curve and a surface that extended into more than two dimensions?
For example, what if $S$ extended vertically in the $z$ direction as $z = 5-x^2-y^2$ and $C$ was defined as the intersection of this paraboloid and the plane $z = x + y$?
Or better yet, what if both $C$ equaled the intersection of that same paraboloid and the sinusoid $z = \sin x + \sin y$, allowing both $C$ and $S$ to extend past two dimensions? 
 A: Let $S1$ and $S2$ be two surfaces bounded by the same closed curve $C$.
Then,
$$\iint_{S1} \vec{B} \cdot d\vec{A} - \iint_{S2} \vec{B} \cdot d\vec{A}$$
$$=\iint_{S1 - S2} \vec{B} \cdot d\vec{A}$$
But, $S1-S2$ forms a closed surface and since the divergence of $\vec{B}$ is zero, this integral vanishes by Gauss' theorem. Hence,
$$\iint_{S1} \vec{B} \cdot d\vec{A}  =\iint_{S2} \vec{B} \cdot d\vec{A}$$
In other words, given a closed curve $C$ you can choose any bounding surface $S$ to compute for Faraday's law according to your convenience. And yes it does hold for any closed curve $C$ even if it is wavy as in your example.
A: What you need is a closed loop $C$ which defines the throat of an open surface $S$.  

The right hand diagram, with the surface $S$ a plane, might be the one which is more commonly used simply because it probably makes any integrations easier.  
Faraday counted magnetic field lines which entered the throat defined by the loop and came out of the other side of the surface which I think is a good way of describing what one is doing when eventuating the magnetic flux (= flow of magnetic field lines) linked to a surface.  
So the closed loop and the surface can be three dimensional.
