1- What kind of advantages does one have by having a canonical symplectic form on $T^*M$ apart from the form being exact? Would it for instance provide any advantage to studying Hamiltonian dynamics of a function using a canonical symplectic form? The only thing that I can think of is that any coordinate change of $T^*M$ induced by a coordinate change of $M$ preserves the symplectic structure which means that the usual coordinates for $T^*M$ are symplectic coordinates.
2- The tangent bundle of a manifold can be given a symplectic structure with a symplectic form $\omega$. We can also construct an atlas where $\omega$ will locally have the form $\sum_i dx^i\wedge dv^i$ and now if we have a function $H: TM \rightarrow R$ we can also find a vector field $X_H$ which satisfies $\omega(X_H,\cdot) = dH$ and the form of the equations will be the same as the Hamilton equations of motion in these coordinates. The only thing is that the natural coordinates on $TM$ will not be the symplectic atlas while the natural coordinates on $T^*M$ can be taken to be the symplectic atlas. I am not sure however how this helps regarding physical questions.
Thanks