How is spacetime depicted in quantum field theory? Is space and time completely separate, and time is just nature of law as in Newtonian mechanics?
The answer here depends on the formalism of quantum mechanics you use. If you use a Hamiltonian formalism, you are picking out some slicing of time, and the Hamiltonian pushes you forward in time. In this formulation, the state-vector is evolving in time (or the operators, whichever way you like to say it), and the description is like Newton's philosophically. The result is relativistically invariant, so it is good to find a relativistic description.
In the path-integral, you describe the quantum state evolution completely invariantly, by a sum over all field histories at intemediate times. In this formulation, the description is manifestly covariant--- the Lagrangian is relativistically invariant, and the state is just given by boundary conditions on the path integral (or by inserting operators to make states).
The description of the state is then the only global non-covariant thing, and this state specification can be done maximally covariantly in Schwinger's way, by saying what the state is on an initial and final space-like hypersurface (a surface any two points of which are spacelike separated). This point of view is the quantum action principle, the Feynman path integral between two spacelike hypersurfaces that specify the initial and final state.
Specifying a quantum state is always a global affair, because the quantum state is a sum over the classical configurations with complex weights for each one. The configurations are global, so you need to consider many different possible configurations at one "time", and this is the role of the space-like surface. But the space-like surface can be moved around arbitrarily, and you can still specify the same state, so it's still obviously relativistic.
The main issue is the standard quantum mechanics thing that you aren't interested only in unitary evolution, but also in measurements, which happen at some future time. The measurements are not naturally included in the formalism for time-evolution, and how you include them is up to you. But the measurement issue is not peculiar to quantum fields. If you take an Everett interpretation of some kind, the path integral evolution is all the mathematics, really, and the interpretation is just a philosophical layer on top which maps different branches of the wavefunction to different experiences of the observers. But you can prune some other way, consistent with some other philosophy.
The more interesting, and in hindsight more fruitful, approach is the Feynman way, inspired by Wheeler's S-matrix, where you take the surfaces to infinity, and consider states of finitely many incoming particles transforming into outgoing particles. This is an interesting point of view, because you can reformulate the quantum fields as point particle motion in a point-particle path integral, which turns out to be equivalent perturbatively to Schwinger and Stueckelberg's covariant quantum field theory when all is said and done. The point-particle formulation generalizes to strings, which do not have a Schwinger picture, since string theory isn't local fields.
In nonrelativistic quantum field theory, space and time are described separately, as in the Newtonian mechanics of fluids. Fields are functions of space and time, and field equations are first order in time. The only change is that fields are operator-valued distributions rather than real-valued functions.
In relativistic quantum field theory, space and time are described jointly, as in special relativity. Fields are functions of spacetime, and field equations are second-order in time for bosonic fields, Dirac-type for fermionic fields. Again the only change is that fields are operator-valued distributions rather than real-valued functions.