Collection of histories vs. collection of momentary configurations For a given Hamiltonian, is the space of histories of a classical system the same as the symplectic manifold? 
Do I have to take care of gauge equivalences and if so, is this only an issue for fields (not for trajectories)?
 A: Basically, the answer to both questions is yes. In the field theory case, however, this is true (optimistically speaking according to Witten-1) - in the case when the field equations are hyperbolic wave equations. In this case one needs the initial conditions to be defined on global Cauchy hypersurface. In this case there is a one to one correspondence between the space of solutions and the space of initial data.
Actually, we always construct the symplectic structure from the Lagrangian in the course of the procedure of canonical quantization, when we construct the conjugate momenta and declare canonical Poisson brackets between the coordinates and their momenta. As a matter of fact, when the coordinates define an affine space (or superspace) like in the theory of spinless point particles or the case of scalar, spinor or Yang-Mills fields, then the phase space is an affine symplectic vector (super)space.
Now, your second assertion is also true (please see Witten-1 and: Witten-2 ). In the presence of gauge freedom, (which is a choice or a declaration that we make); the space of solutions modulo the gauge freedom is symplectic and is identified with the physical phase space of the theory. This identification is natural because we measure only gauge invariant quantities. The first assertion that this space is symplectic stems from a deep theorem in symplectic geometry called the Marsden-Weinstein reduction. (This theorem is true in finite dimensions but was generalized to many infinite dimensional cases), which asserts that the reduction of a symplectic manifold by a gauge freedom is also symplectic. 
Another way to look at the gauge redundancy is to observe that there are certain functions of the initial data that can be arbitrarily evolved in time and still the field equations are satisfied, please see Belot.
Now, in contrast to the unreduced symplectic manifolds which are in our cases nice symplectic vector spaces, the reduced manifolds are actually
orbifolds. These spaces contain singular submanifold which may be classically harmless as long as one stays away from them, but have profound effect after quantization, which is not very well understood until now.
There are cases when the unreduced phase spaces of certain gauge theories are infinite dimensional while the reduced ones are finite dimensional.
These cases led Witten to one of his greatest achievements, namely in the Chern-Simons theory or in the 2+1 dimensional gravity theory in Witten-2.
