EDIT: I noticed that your book is not using the usual set-theoretic definition of natural numbers, where $n = \{0,1,2,\ldots, n-1\}$. Namely, $0$ is not considered to be a natural number, so the prototypical $n$-element set is the set $\{1,\ldots,n\}$ rather than $n$ itself. I will update my answer to reflect this.
I think you are making this more complicated that it needs to be. The notion of countability does not seem to be relevant. As you said, we define a set $x$ to be finite if there is a bijection between $x$ and some natural number $n \in \mathbb{N}$ (or it is empty.) Let $x$ be a nonempty set.
If $x$ is finite, then there is a bijection from $x$ onto a finite set $y$: namely, let $y=x$ and consider the identity function.
Conversely, assume there is a bijection $f$ from $x$ onto a finite set $y$. By definition of "finite" there is a bijection $g$ from $y$ onto the set $\{1,\ldots,n\}$ for some natural number $n$. You can then show that the composition $g \circ f$ is a bijection from $x$ onto the set $\{1,\ldots,n\}$, so $x$ is finite as well.
One problem with your argument for the forward direction is that the theorem you are applying deals with countable sets rather than finite sets, so you cannot use it to conclude that there is a bijection from anything to a finite set. In fact, the theorem is irrelevant. I didn't finish reading the rest of this direction.
Your argument for the reverse direction is vague. You are proving that something is finite, so you should mention some natural number $n$, some set of the form $\{1,\ldots,n\}$, and some bijection between this set and some other set.
Your proof is correct, but the statement you are proving is not the same as "$\aleph_0$ is the smallest cardinality of infinite sets". You have proved that there is no cardinality of infinite sets that is smaller than $\aleph_0$. However, there may exist a cardinality of infinite sets that is incomparable with $\aleph_0$. In other words, there could exist a set $A$ such that there does not exist an injection $A\to\mathbb{N}$ and there also does not exist an injection $\mathbb{N}\to A$.
Best Answer
Let $f:A \to B$ be a bijection and define $F:\wp(A) \to \wp(B)$ by
$F(M):=f(M)$.
Then $F$ is a bijection.