What is the number of extensive variables needed to characterize a system? How is it that in the absence of an external field a system can be characterized by its energy, volume, and the number of particles of each species? Why is this number sufficient?
 A: In general, this is not true. In the absence of external coupling, a system can be "characterized" by the values of all conserved quantities*. For the kinds of systems studied in introductory thermodynamics classes, energy, volume, and number density are usually assumed to be the only relevant conserved quantities*.
*I'm including parameters of the Hamiltonian (e.g. volume of a box) in the set of conserved quantities. These parameters don't change over time (by assumption), so they are mathematically equivalent to extra conserved quantities in a system with corresponding extra degrees of freedom.
Now, what does "characterized" mean? It means that the long-run average values of all observable quantities are fully determined by the values of the conserved quantities. Under some conditions, this also implies that the large-$N$ behavior of the system is fully determined by the values of the conserved quantities.
If we're willing to be careless with edge cases, this statement is not so hard to prove. The long-run average value of an observable $f(q,p)$ is given by $$\langle f \rangle = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T}  f(q(t),p(t)) dt$$
We can split the phase space up into the different trajectories of the system. Once we take the limit $T \rightarrow \infty$, the long-run average shouldn't depend on when along the trajectory we started. It should depend only on which trajectory we are on. But which trajectory we are on is fully determined by the values of any complete set of conserved quantities. We can see this by defining a special conserved quantity for each path of the system which is $1$ along the path and $0$ elsewhere.
