Stokes / Green's theorem with non-regular regions

Question

One statement of Green's Theorem (Stewart) I have seen is:

Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$. If $\mathbf{F}$ has continuous partial derivatives on an open region that contains $D$, then $$\oint_C \mathbf{F} \cdot d \mathbf{\ell} = \int_D \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) dA$$

Whereas another statement I have seen is:

Let $D$ be a regular region with a positively oriented, piecewise-smooth boundary $C$. If $\mathbf{F}$ has continuous partial derivatives on an open region that contains $D$, then $$\oint_C \mathbf{F} \cdot d \mathbf{\ell} = \int_D \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) dA$$

A regular region is a compact subset of $\mathbb{R}^n$ (closed and bounded) which is the closure of its interior ("no isolated points")

These two definitions are similar but are slightly different. Note that the second replaces the "$C$ is simple closed curve" with "$D$ is a regular region". I know that there is a lemma stating:

A regular region $D \subseteq \mathbb{R}^n$ has a piecewise smooth boundary if its boundary $S = \partial D$ is a finite union of piecewise, simple closed curves.

That lemma seems to connect the two, except that:

(a) the second definition of Green's theorem still has the "bounded" condition that the first doesn't.

(b) the lemma goes one way, i.e. if the boundary $S$ is a finite union of piecewise, simple closed curves, then the regular region $D$ has a piecewise smooth boundary. But supposedly then a regular region $D$ could still have a piecewise smooth boundary even if the boundary is not a finite union of piecewise, simple closed curves?

My questions then are:

1. Is the "bounded" constraint necessary? i.e. can we still apply Green's or Stokes' theorem to unbounded curves / regions? The closest thing I could find to answer this was in this post, but that only gives the answer of "maybe" using the example of a vector field that goes to zero at infinity.

2. Do we gain anything by using the notion of regular regions in the definition rather than defining the region with simple closed curves? What sort of more `general' curves is this allowing?