This is only a **long comment**.

We clearly have $i(D) = D \setminus \{ 0 \}$ for any Bézout domain $D$, while $i(k[x, y, z]) = k \setminus \{ 0 \}$ by Krull's principal ideal theorem.

As exemplified by js21 with $D = k[x, y]$ in a comment above, the answer to OP's first question is **no in general**. Still, I'll provide a small glint of hope.

OP's definition can be rephrased as follows.
>> **Rewording.** The set $i(D)$ consists of the elements $a \in D \setminus \{ 0 \}$ such that $D/Da$ is a principal ideal ring (PIR).

If $a, b \in i(D)$ are coprime, e.g., $Da + Db = R$, then we have have
$D/Dab \simeq D/Da \times D/Db$ by the Chinese Remainder Theorem. Thus $ab \in i(D)$. In particular the set of primes $p \in D$ such that $Dp$ is a maximal ideal of $D$ is a submonoid of $D \setminus \{0\}$ contained in $i(D)$.

Using the above rewording, we can quickly check why js21's comment answers the question in the negative. If $k$ is field, we have $k[x, y]/(x) \simeq k[y]$ which is a principal ideal domain whereas $k[x, y]/(x^2) \simeq (k[x]/(x^2))[y]$ is not a PIR (the ideal generated the images of $x$ and $y$ is readily seen not to be principal). 
If $k$ is algebraically closed then $i(k[x, y]) = \{\alpha x + \beta, \alpha' y + \beta' \, \vert \, \alpha, \alpha', \beta, \beta' \in k\} \setminus \{ 0 \}$, which is trivially saturated.

Let us give another example for which multiplicative closedness of $i(D)$ fails dramatically. Consider a regular local ring $D$ (this is necessarily a domain) of Krull dimension $2$ with maximal ideal $\mathfrak{m}$. Then any lift in $\mathfrak{m}$ of a non-zero element of $\mathfrak{m}/\mathfrak{m}^2$ belongs to $i(D)$, but any product of two such lifts doesn't. (The set $i(D)$ is trivially saturated).


**Side note**. The answers to this [MSE question][1] show why $i(D) = D \setminus \{0\}$ when $D$ is a Dedekind domain and point to interesting references.


  [1]: https://math.stackexchange.com/questions/597543/in-a-dedekind-domain-every-ideal-is-either-principal-or-generated-by-two-element