Complements of Simply Connected Subsets of the Plane

Moore's theorem says that if $\sim$ is an equivalence relation on $\mathbb{S}^2$ such that any equivalence class is closed connected and has connected complement then the quotient space $\mathbb{S}^2/\sim$ is homeomorphic to $\mathbb{S}^2$.

In particular the answer to your first question is "yes".

After a year the question may be "cooled"; on the other hand good questions never cool off.

Your question concerns the relationship between continua $X \subset \mathbb{R}^2$ amd their complements $CX = \mathbb{R}^2 -X$.

Shape theory provides an answer. We have $CX \approx CY$ iff $X$ and $Y$ have the same shape. There are countably many shapes of continua $X \subset \mathbb{R}^2$: These are represented by $X_0$ = one-point-space, $X_n =$ wedge of $n$ circles and $X_\infty =$ Hawaiin earring. Therefore $CX \approx \mathbb{A}$ iff $X$ has trivial shape (i.e. the shape of $X_0$).

In particular, $CX \approx \mathbb{A}$ iff $CX$ is connected.