In hindsight, Noether's theorem is a dramatic hint of quantum mechanics. Mariano is completely correct in his comment that the conserved quantity is $A$ itself, but it deserves a bit of explanation.
A classical probabilistic system is characterized by an algebra of random variables. You could consider the Boolean random variables, in which case the algebra is a $\sigma$-algebra $\Omega$. Or you could consider real or complex random variables; if you take the bounded ones then the algebra is $L^\infty(\Omega)$. In quantum probability, you have the same sort of thing, except that the algebra of bounded complex random variables is a non-commutative von Neumann algebra. One choice with special properties is the algebra $\mathcal{B}(\mathcal{H})$ of all bound operators on a Hilbert space $\mathcal{H}$.
The special property of $\mathcal{B}(\mathcal{H})$ is that all automorphisms are inner, so that any symmetry $A$ of a quantum dynamical system is necessarily also a random variable that you can measure. This does not happen classically, nor even for other non-commutative von Neumann algebras. Even without writing down a time-independent Schrodinger equation, it makes Noether's theorem trivial, because the symmetry $A$ must be conserved if you interpret it as a quantum random variable. Unlike in the classical case, $A$ doesn't even need to generate or come from a continuous group action.
For example, the parity operator (which negates all three coordinates of space) is a conserved quantity of electromagnetism, so it leads to a (two-valued) conserved quantity in quantum electrodynamics which is also called parity. The discrete symmetry also exists classically as a symmetry of Maxwell's equations (if you are careful to negate the magnetic field vectors twice), but the classical Noether's theorem doesn't apply.
Anyway, the identity operator is the trivial random variable that is always 1, as Aaron says.
Your Question 1 is called Darboux theorem for fibrations (see: Arnold, V., Givental, A., Symplectic geometry, Dynamical Systems IV, Symplectic Geometry and its Applications (Arnold, V., Novikov, S., eds.), Encyclopaedia of Math. Sciences 4, Springer-Verlag, Berlin-New York, 1990.)
Here is how to construct suitable Darboux coordinates. Let $q_i$ be local coordinates in the base of the fibration, we identify them with their pullbacks to the symplectic manifold. The functions $q_i$ generate Hamiltonian vector fields $X_{i}$ and these fields are tangent to the fibers (note that $X_{i}$'s commute). Let $\varphi_{i}(t)$ be the flow map generated by $X_{i}$ for time $[0,t]$.
Now we choose (locally) a Lagrangian submanifold $L$ transversal to the fibration. The coordinates $q_i$ give coordinates on $L$, so $(q_1,...,q_n)$ stands for a point on $L$. Here is a construction of a local symplectomorphism $$(p_1,...,p_n,q_1,...,q_n) \mapsto \varphi_{n}(p_n)\circ ...\circ \varphi_{1}(p_1)(q_1,...,q_n).$$ It is easy to check that it is indeed a fibered symplectomorfism sending the symplectic structure to the standard one.
Best Answer
This echos the 2017 comments, but since the question has now been bumped to the front page it might be helpful to give the actual source in Maslov's book [1].
[1] V.P. Maslov, Perturbation Theory and Asymptotic Methods (in Russian, Moscow, 1965); Théorie des pertubations et méthodes asymptotique (French translation, Paris, 1972).
The Lagrangian manifold (variété lagrangienne) is introduced on page 114-115: