A ring such that $(a+b)^2=a^2+b^2$ and $(a+b)^3=a^3+b^3$

This is the same, but maybe rearranged to see an idea of norming (non-commutative) monomials in $a,b$. As noticed in the OP, from $a^2+b^2=(a+b)^2=(a+b)(a+b)=a^2+ab+ba+b^2$ we get the "supercommutativity relation" $$ ab= -ba\ . $$ The other relation, extended as $a^3+b^3=(a+b)^3=(a+b)^2(a+b)=(a^2+b^2)(a+b)=a^3+a^2b+b^2a+b^3$ gives as in the OP $$ a^2b=-b^2a\ . $$

We start now a "new" proof. (Essentially the same.) The strategy being to write each monomial in $a,b$ in a "normalized form". First, using supercommutativity we can write any monomial $aa\dots abb\dots baa\dots a bb\dots b\dots$ in the form $\pm aa\dots a\ aa\dots a\ \dots bb\dots b\ bb\dots b\dots$ by pushing all $a$'s in the front and changing signs. Using the second rule, we can reduce the $a$ powers in front of the $b$'s to get a "normalized" monomial of the shape $\pm b^?a$. Let us show inductively the relation $a^nb=-b^na$, starting from the given one for $n=2$. For $n\ge 2$ we have $$ \begin{aligned} a^{n+1}b &= aa^n b\\ &= -ab^n a&&\text{(by induction)}\\ &= -(-1)^n aa b^n&&\text{(by supercommutativity)}\\ &=-(-1)^n a^2 \underbrace{bbb\dots b}_{n\text{ times}}\\ &=-(-1)^n(- b^2 a)\underbrace{bb\dots b}_{n-1\text{ times}}\\ &=+(-1)^n b^2\ (a\underbrace{bb\dots b}_{n-1\text{ times}})\\ &=+(-1)^n b^2\ (-1)^{n-1}\underbrace{bb\dots b}_{n-1\text{ times}}a&&\text{(by supercommutativity)}\\ &=-b^{n+1}a\ . \end{aligned} $$ The wanted relation now follows also inductively, $$ (a+b)^{n+1}=(a^n+b^n)(a+b)=a^{n+1}+\underbrace{a^nb+b^na}_{=0\text{ shown above}}+b^{n+1} = a^{n+1}+b^{n+1} \ .$$

It is no hard to see by using induction $a^nb=-b^na$ and $a^nb^n=-b^na^n$

Assume that $n$ is even. $$(a+b)^n= \underbrace{(a+b)^2(a+b)^2(a+b)^2(a+b) ^2\cdots(a+b)^2}_{\text{$\frac{n}{2}$}\ \ \text{times}}$$
$$=(a^2+b^2)(a^2+b^2)\cdots(a^2+b^2)$$ $$=(a^4+a^2b^2+b^2a^2+b^4)\cdots (a^4+a^2b^2+b^2a^2+b^4)$$ Using $a^nb^n=-b^na^n$ $$=(a^4+b^4)(a^4+b^4)\cdots(a^4+b^4)$$ $$\vdots$$

• Finally, we reach two components by using $a^nb^n=-b^na^n$, We get $a^nb=-b^na$ if used as much as necessary we obtain $$=a^n+b^n$$ $\textbf{Note:}$ When we reach the odd number of components, we do not handle the final component.

$\textbf{Example:}$ The logic is as follows: $$(a+b)^6=(a+b)^2(a+b)^2(a+b)^2$$ $$=(a^2+b^2)(a^2+b^2)(a^2+b^2)$$ $$=(a^4+a^2b^2+b^2a^2+b^4)(a^2+b^2)$$ $$=(a^4+b^4)(a^2+b^2)=(a^6+a^4b^2+b^4a^2+b^6)$$ $$=a^6-b^4ab-a^4ba+b^6$$ $$=a^6+b^5a+a^5b+b^6=a^6+b^6$$ Similarly, we can think when $n$ is odd as following:(Actually, we encounterd above the situation below.) $$(a+b)^n= \underbrace{(a+b)^2(a+b)^2(a+b)^2(a+b)^2\cdots (a+b)^2}_{\text{$\frac{n-1}{2}, $}\ \ \text{times}} \underbrace{(a+b)}_{\text{$1$ times}}=a^n+b^n $$