State, Dynamics and Interpretation in QFT In physical theories that are farther away from our everyday experience, part of the difficulty, apart from the math by itself, lies in grasping the physical meaning of the theory. 
In quantum mechanics, I think I only managed to start to make sense of things when I clearly saw the following three separate constituents as separate ingredients:


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*Physical state, as an abstract element of a given Hilbert (state) space, or maybe a more concrete representation as an $L^2$ space of functions. This is the true objective physical reality, without necessarily assigning an a priori physical meaning to it yet.

*Dynamics: these are the laws of physics, of time evolution and interaction. They allow us to make predictions about states. In this case it is the Schrödinger or the Heisenberg equation of motion which exactly prescribes how a physical state evolves in time. Even though I really like the elegance of the Heisenberg picture, I think that for the clear distinction of state and evolution the Schrödinger picture may be more appropriate.

*Interpretation: this is the step in which meaning is assigned to physical states. Though we have our objective physical states and their behaviour over time and through interactions from the previous two points, we still need a way to link this to the physical world we see around us. In quantum mechanics this would be the Born rule, which to a given quantum state associates probabilities of outcomes of measurements.
In different contexts it may be useful to start mixing these ingredients up in different ways or abstracting them away, like in the Heisenberg picture, where the first two (an in a way the third as well) are mostly intertwined, in quantum computation where the dynamics are essentially fully abstracted away into quantum circuits while retaining state and interpretation/measurement in a very essential way, and in the path integral formalism you mostly abandon the actual physical state and restrict to classical states and transition probabilities, thus retaining only the last two. Even then, I think that mentally having access to these three clearly distinguished ingredients as listed gives a very useful framework to fall back onto in case of confusion.
In quantum field theory I'd like to have a similar mental framework, but I didn't really manage so far. To start, I guess the physical state would be a classical (scalar, vector, spinor) field. We usually work with fields of operators, but I guess we really are in the Heisenberg picture then, which in the treatments of QFT that I have seen so far seems to be a much more fundamentally useful or necessary way of looking at things than in quantum mechanics, where it is essentially an alternative way to (mathematically) look at something that was well-defined already. 
Then the few things that we can compute (scattering, decay) are done by perturbation theory to partially evaluate the Dyson series. In this process, I feel that state, dynamics and interpretation are intertwined in a very intricate way, that doesn't help me in forming the mental framework that would help me order my understanding.
Is there any way to think about state, dynamics and interpretation in a way similar to quantum mechanics? In particular, with a state that is essentially a classical field, and dynamics that can be formulated independent of observables or outcomes of measurements? 
If this question doesn't make sense, if you can explain me why that will also without a doubt be very enlightening. 
 A: 
Is there any way to think about state, dynamics and interpretation in a way similar to quantum mechanics? 

Yes. The answer to all of these questions in quantum field theory is exactly the same as in quantum mechanics, because quantum field theory is merely a special case of quantum mechanics. In Schrodinger picture, the state of the system is given by a vector in a Hilbert space, and the time evolution is given by the Schrodinger equation.
More specifically, a classical particle can have any position $x$, so the state is
$$|\psi\rangle = \int dx \, \psi(x) |x \rangle.$$
Similarly, a classical field can have any field values $\varphi(x)$, so the state is
$$|\Psi \rangle = \int \mathcal{D} \varphi \, \Psi(\varphi) |\varphi(x) \rangle.$$
This is called the Schrodinger functional picture of QFT. As you said, this isn't very convenient, so relativistic QFT is just about always done focusing on the Heisenberg picture. 

Then the few things that we can compute (scattering, decay) are done by perturbation theory to partially evaluate the Dyson series. 

Sure, there's a ton of machinery here (LSZ reduction, Wick's theorem, the Gell-Mann Low theorem, cross section formulas) but it's mostly a red herring. We care a lot about $S$-matrix elements because it's hard for colliders to measure anything else. But in condensed matter field theory one is often interested in calculating, say, the spectrum of the Hamiltonian, or position-space correlation functions. These don't require the cross-section machinery above and are totally familiar from quantum mechanics. There's no fundamental difference between QM and QFT.
