Here is some background on adjoining elements from Artin:

I'm not sure I understand the "dropping bars" part. Why can we drop the bars?

Here is a theorem following this discussion:

• Red part: Why do elements of $R[\alpha]$ have the form $g(\alpha)$? It is certainly clear if I use the definition of $R[\alpha]$ saying that this is the smallest ring containing $R$ and $\alpha$. But in this context $R[\alpha]$ is a name for the ring $R[x]/(f)$, and don't understand why elements of $R[x]/(f)$ are polynomials in $\alpha$. The elements of $R[x]/(f)$ are, in my understanding, polynomials of the form $\sum \overline a_i \overline x^i=\sum \overline a_i \alpha^i$, and again, if we can "drop bars", this is just $\sum a_i \alpha^i$, and my question is answered, but why can we do that?
• Blue part: How does uniqueness follow from this?
• Green part: I don't understand why Artin writes "isomorphic" here. Doesn't he define $\mathbb Z[\sqrt[3] 2]$ to be $\mathbb Z [x]/(x^3-2)$? I'm also not sure why Artin refers to that substitution map and mentions the kernel instead of just saying "consider $\mathbb Z[\sqrt[3] 2]$".

First picture: This is just some notational laziness/efficiency. You're correct that technically we should define $\overline{f} \in (R[x]/(f))[y]$ as $\overline{a_n} y^n + \cdots + \overline{a_0}$, and say $\overline{f}(\alpha) = 0$, but this is tiresome to write out. Moreover, in nice situations, for instance if $R$ is a domain, then $\pi: R[x] \to R[x]/(f)$ maps $R$ isomorphically onto its image if $f$ is nonconstant. In many applications $R = F$ is a field, and once we form $K = F[x]/(f)$ we identify $F$ with its image under the quotient map so that we can say that $F$ is a subfield of $K$. For instance, if we define $\mathbb{C} = \mathbb{R}[x]/(x^2+1)$, then $\mathbb{R}$ isn't literally a subfield (or even a subset) of $\mathbb{C}$, but is isomorphic to the subfield $\{\overline{r} : r \in \mathbb{R}\}$.
(i) Yes, this is the same "dropping bars" game. We write $a_i$ for $\overline{a_i}$ and just remember that we are working in the quotient.
(ii) He's basically just using the fact that remainders are unique in the division algorithm. If $\beta = r_1(\alpha)$ and $\beta = r_2(\alpha)$, where $r_1$ and $r_2$ both have degree $< n$, then $0 = r_1(\alpha) - r_2(\alpha)$, so $r_1(x) - r_2(x) \in (f)$. But if $r_1(x) - r_2(x)$ is nonzero then it has degree $< n$, contrary to the lemma. Thus $r_1 = r_2$ and the expression for $\beta$ is unique.
(iii) Sure, that is how the notation was defined in the proposition. I think here Artin is considering $\mathbb{Z}[\sqrt[3]{2}]$ as the smallest subring of $\mathbb{C}$ containing $\mathbb{Z}$ and $\sqrt[3]{2}$, as you mentioned above. If I recall correctly, earlier in the book Artin defines $$\mathbb{Z}[\sqrt[3]{2}] = \{a + b \sqrt[3]{2} + c \sqrt[3]{2}^2 : a, b, c \in \mathbb{Z}\}$$ so that is another possible interpretation.