If I have the distinct points $(x_{1}, 0) ... (x_{n}, 0)$,
A) What would be a simple polynomial (non-constant) passing through these points? And could I instead also write a simple continuous curve (a function) using the sine function?
B) Intuitively, I would expect that, by necessity, such a (continuous) polynomial or wave must change sign at each point, in order to be able to go through all points (without being a constant). How can this intuition be justified formally?
C) Imagining that the set of points becomes arbitrarily large (and they are equispaced, as a further simplification), is it true that the polynomial converges to a definite curve (for instance, a sinusoidal)? And why?
Regarding your last question, is it indeed true (but not obvious) that
$$ \frac{\sin x}{\ x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right) =\prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi}\right)\left(1+\frac{x}{n\pi}\right) $$
As $\frac{\sin x}{ x}$ has infinitely many roots $\pm\pi, \pm2\pi, \pm3\pi ...$ and it can be seen as a limiting product of an interpolating polynomial with an arbitrary large degree.