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 $\in[0,t]$. [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 syplectomorfism symplectomorfism sending the symplectic structure to the standard one. 1 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$\in[0,t]$. 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 syplectomorfism sending the symplectic structure to the standard one.