Yes. The generalization is provided by modular arithmetic. The properties you are observing all come from the fact that taking the remainder modulo $n$ respects addition and multiplication, and this generalizes to any $n$. More generally in abstract algebra we study rings and their ideals for the same reasons.
The notion of evenness and oddness of functions is closely related, but it is somewhat hard to explain exactly why. The key point is that there is a certain group, the cyclic group $C_2$ of order $2$, which is behind both concepts. For now, note that the product of two even functions is even, the product of an even and odd function is odd, and the product of two odd functions is even, so even and odd functions under multiplication behave exactly the same way as even and odd numbers under addition.
There are also huge generalizations depending on exactly what you're looking at, so it's hard to give a complete list here. You mentioned chessboards; there is a more general construction here, but it is somewhat hard to explain and there are no good elementary references that I know of. Once you learn some modular arithmetic, here is the modular arithmetic explanation of the chessboard idea: you can assign integer coordinates $(x, y)$ to each square (for example the coordinate of the lower left corner), and then you partition them into black or white squares depending on whether $x + y$ is even or odd; that is, depending on the value of $x + y \bmod 2$. Then given two points $(a, b)$ and $(c, d)$ you can consider the difference $c + d - a - b \bmod 2$, and constraints on this difference translate to constraints on the movement of certain pieces. This idea can be used, for example, to prove that certain chessboards (with pieces cut out of them) cannot be tiled with $1 \times 2$ or $2 \times 1$ tiles because these tiles must cover both a white square and a black square. Of course there are generalizations with $2$ replaced by a larger modulus and larger tiles.
As for matrices and vectors, let's just say that there are a lot of things this could mean, and none of them are straightforward generalizations of the above concept.
I wouldn't appeal to the fact that the integers alternate between even and odd.
I would say this:
An even number is defined as any integer of the form $2k$ for $k$ any integer (maybe excluding $k=0$).
An odd number is defined as any integer of the form $2k +1$ for any integer $k$.
You can just take this as the definition. And then (as you have) the proof that even times odd is even is just
$$
2k(2k'+1) = 2[k(2k'+1)]
$$
where of course $k(2k'+1)$ is just an integer.
Best Answer
Yes, it is.
Consider the set $\Bbb N$ or natural numbers and whatever "pattern" $(p_n)_{n \in \Bbb N}$ you want. Define $f(n)=p_n$ and then interpolate linearly in $[n, n + 1].$ Make $f$ constant until the first natural.