Any symplectic linear transformations in $T_xM$ is locally realizable as a Hamiltonian vector field, thus for questions 1 and 2, one can profitably use representation theory of the symplectic group.
FACT (Lefschetz decomposition) Let $W$ be a $2n$-dimensional symplectic vector space, $\bigwedge^\ast W$ its exterior algebra, and $\omega\in\bigwedge^2 W$ the invariant two-form. Exterior multiplication by $\omega$ and the contraction with $\omega$ define a pair of $Sp(W)$-equivariant graded linear transformations $L, \Lambda$ of $\bigwedge^\ast W$ into itself of degrees $2$ and $-2,$ and let $H=\deg-n$ be the graded degree $0$ map acting on $\bigwedge^k$ as multiplication by $k-n.$ Then $L,H,\Lambda$ form the standard basis of the Lie algebra $\mathfrak{sl_2}$ acting on $\bigwedge^\ast W$ and the actions of $Sp(W)$ and $\mathfrak{sl_2}$ are the commutants of each other.
See, for example, Roger Howe, Remarks on classical invariant theory.
Corollary Every homogeneous $Sp(W)$-invariant element of $\bigwedge^\ast W$ is a multiple of $\omega^k$ for some $0\leq k\leq n.$
Since, conversely, every polynomial in $\omega$ is invariant under the Hamiltonian vector fields, this gives a full description of the invariant differential forms.
For question 2, locally every invariant tensor must reduce to an $Sp(W)$-invariant element of the tensor algebra. For the special case of symmetric tensors, the answer is trivial.
FACT Under the same assumptions, the $k$th symmetric power $S^k W$ is a simple $Sp(W)$-module (non-trivial for $k>0$).
General case can be handled using similar considerations from classical invariant theory. A more involved question of describing the invariant local tensor operations on symplectic manifolds (an analogue of the well-known problem of invariant local operations on smooth manifolds, such as the exterior differential or Schoutens bracket) was considered in an old article by A.A.Kirillov.
My reading of the question is this: we're given $H\in C^\infty(M)$ with $M$ symplectic, and we want to know whether there's a submanifold $L\subset M$, a Riemannian metric $g$ on $L$, and a symplectomorphism $T^\ast L \cong M$ under which $H$ pulls back to the norm-square function. And we want to know if $(L,g)$ is unique.
Uniqueness is easy: we recover $L$ as $H^{-1}(0)$, and $g$ as the Hessian form of $H$ on the vertical tangent bundle (determined by the symplectomorphism) along $L$.
Basic necessary conditions:
(1) $L:=H^{-1}(0)$ is a Lagrangian submanifold of $M$.
(2) $L$ is a non-degenerate critical manifold of $H$ of normal Morse index 0.
These conditions imply that a neighbourhood of $L$ embeds symplectically in $T^\ast L$, and also (by the Morse-Bott lemma) that $H$ is quadratic in suitable coordinates near $L$. These two sets of coordinates needn't be compatible, so let's replace (2) by something much stronger (but still intrinsic):
(3) There's a complete, conformally symplectic vector field $X$ (i.e., $\mathcal{L}_X\omega=\omega$), whose zero-set is exactly $L$, along which $H$ increases quadratically (i.e., $dH(X)=2H$).
I claim that (1) and (3) are sufficient. With these data, you can locate a point $x\in M$ in $T^\ast L$. Flow $X$ backwards in time starting at $x$ to obtain the projection to $L$; pay attention to the direction of approach to $L$ to get a tangent ray, and use the metric (i.e., the Hessian of $H$ on the fibres of projection to $L$) to convert it to a cotangent ray. Pick out a cotangent vector in this ray by examining $H(x)$. If I'm not mistaken, this will single out a symplectomorphism with the desired properties.
Best Answer
Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say, ${\bf R}^n$, are (in principle, at least) generated by the position operators $x_j$ and the momentum operators $\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$ \frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where $[,]$ here is the commutator $[A,B] = AB-BA$.
Meanwhile, classical observables on the phase space $T^* {\bf R}^n$ are (again in principle) generated by the position functions $q_j$ and momentum functions $p_j$, which are related to each other by the basic commutation relations $$ \{ q_j, p_k \} = \delta_{jk}$$ where $\{,\}$ is now the Poisson bracket (I may have the sign conventions reversed here).
One of the great insights of quantum mechanics (or, on the mathematical side, semi-classical analysis) is the correspondence principle that asserts, roughly speaking, that the behaviour of quantum observables converges in the high-frequency limit (or, after rescaling, the semi-classical limit) to the analogous behaviour of classical observables. The correspondence is easier to see on the observable side than on the physical space side, for instance by connecting the von Neumann algebra of bounded quantum observables with smooth symbol with the Poisson algebra of smooth classical observables. The former is connected to linear PDE and the latter to symplectic (or Hamiltonian) geometry.
Another way to see the connection is to investigate what happens when one applies a linear partial differential (or pseudodifferential) operator to a high-frequency function (or "quantum state"), when viewing that function through its Wigner transform, which can be viewed as approximately describing the quantum state by a classical one. A standard calculation shows (under Weyl quantisation) that the top order contribution of the operator on this transform is given by its symbol, and the next order term is basically given by the Hamiltonian vector field associated to that symbol. (This is discussed for instance in Folland's "Harmonic analysis on phase space".) This suggests that the dynamics of linear PDE at high frequencies are going to be driven by the associated Hamiltonian dynamics of the symbol of that PDE.