I know the conservation of energy comes from Noether's theorem via the time-translational symmetry, and if I remember correctly, the conservation of momentum comes from space-translational symmetry.
My question is: for a given system, what is the starting point of identifying conservation laws, and how do you know that there's not more that you haven't identified? Do you just start by identifying symmetries?
As an example, consider a system of hard, spherical, elastically colliding billiard balls. I suppose the conserved quantities are energy, momentum, and angular momentum; for each particle these correspond to the terms $\frac{1}{2}mv^2$, $mv$, and $m\omega$. How do we know there isn't something like $kmv^3$? (I know this isn't one, but it's just to demonstrate my point).
And why is the conserved quantity always a scalar? Are there cases where the conserved quantity is not calculated by summing other scalars?