An idea with pedagogical importance here is that students often have trouble distinguishing between objects that look identical but are mathematically distinct, and these frequently appear in school algebra when students work with polynomial expressions. These expressions can represent (1) expressions involving arbitrary numbers, (2) formal expressions, and (3) functions. Mathematicians commonly refer to objects of the second type as polynomials and objects of the third type as polynomial functions.
As one example, Steven Gubkin brought up how students might think that the polynomial $x^2+1$ is prime/irreducible. This is true if we consider $x^2+1$ as an element of $\mathbb{R}[x]$, the ring of polynomials in $x$ that have real coefficients, but it is not true if we consider it as an element of $\mathbb{C}[x]$, the ring of polynomials in $x$ that have complex coefficients.
I'll first address your questions about case (2) and case (3) by carefully distinguishing between polynomials and expressions involving arbitrary integers. Anyone who is only interested in the pedagogical discussion can skip the next section.
In your cases (2) and (3), the expressions $a+b$, $a^3+b^3$, and $a^2+ab+b^2$ represent both
- Polynomials with indeterminate $a$ and constant $b$ (an element of the underlying ring), and
- Integers given two arbitrary integers $a$ and $b$.
To help distinguish these, instead of $a$ and $b$, I'll use $X$ and $c$ when talking about polynomials, and I'll use $n$ and $m$ when talking about integers.
The factor theorem is a theorem about polynomials. It states that $X-r$ is a factor of the polynomial $P(X)$ if and only if $r$ is a root of $P(X)$, i.e. $P(r)=0$.
Given the polynomial $P(X)=X^3+c^3$, since $P(-c)=0$, we know that $X+c$ is a factor of $X^3+c^3$. This means that $X^3+c^3$ is the product of $X+c$ and some polynomial $Q(X)$. Specifically,
$$(X+c)(X^2-cX+c^2)=X^3+c^3.$$
So far, no integers have been involved. Now, if we replace $X$ and $c$ with arbitrary integers $n$ and $m$, then we get a general relationship between integers,
$$ (n+m)(n^2-nm+m^2) = n^3 + m^3, $$
which we could also call an identity. This identity tells us that, for any integers $n$ and $m$, $n^3+m^3$ is the product of $n+m$ and some integer, and therefore $n+m$ divides $n^3 + m^3.$
Sometimes zero is not allowed as a divisor, in which case we can restrict our conclusion that $n+m$ divides $n^3+m^3$ to integers $n$ and $m$ such that $n+m\neq0$. There's no need for a similar restriction on the statement that $X+c$ divides $X^3+c^3$, because $X+c$ is not the zero polynomial. However if $n$ and $-m$ have the same value, then $n+m$ is the integer zero.
To your question about case (2), notice that nowhere in this argument did we assume that $n$ and $m$ have different signs. We took a relationship between polynomials that the factor theorem helped us to establish, and we used it to deduce a general relationship between integers. If we wanted to, we could restrict it to a general relationship between positive integers: for any positive integers $n$ and $m$, $n+m$ divides $n^3+m^3$.
In case (3), we are asked to find specific relationships between integers. The factor theorem tells us that, as polynomials, $X+c$ is not a factor of $X^2+cX+c^2$. This means that we cannot use the same argument we used above to deduce a general relationship/identity involving integers. However, there may be non-general integer relationships that have this form. And indeed,
$2+2$ divides $2^2+2\cdot2+2^2.$
There's a conversation to be had about when students should be introduced to the level of abstraction at which polynomials and expressions involving arbitrary numbers are seen as distinct objects. I think expressions like $a^3+b^3$ should first be introduced in the context of generalized arithmetic and not formal algebra or functions, which Hung-Hsi Wu makes an argument for in this article. But also, prior to introducing the factor theorem in his textbook Algebra and Geometry, Wu says,
Up to this point, the algebra we have been doing may be called generalized arithmetic, because the symbols employed in all the expressions we have come across all stand for numbers. It follows that all our computations have been with numbers. This kind of algebra is more or less the algebra of the period from al-Khwarizmi (780–850) to 1800. After 1800, the work of Gauss, Abel, Galois, and others slowly transformed algebra into an abstract study of structure. School algebra, at some point, must also give an introduction to such abstract considerations. The study of polynomials provides the most natural venue to showcase the transition from algebra as generalized arithmetic to formal algebra. (p. 175).
I would argue that students need to be aware of this transition in order to flexibly solve problems and construct arguments using the factor theorem, and in turn, the factor theorem motivates the transition. At the secondary level, students don't need the machinery and language of polynomial rings, but they need to have a sense that the statement "the polynomial $x+1$ is a factor of the polynomial $x^3+x+2$" is of a completely different kind than the statement "given an integer $a$, $a+1$ is a factor of $a^3+a+2$."
