Differences between principles of QM and QFT
All knowable information about a system is encoded in a ray in a Hilbert space. In QFT, and unlike non-relativistic QM, there is no $|x\rangle$ basis, so you cannot construct a wave-function $\varphi(t,x)=\langle x|\varphi(t)\rangle$ to encode this information. What you can do is encode this information in the so-called correlation functions (cf. Wightman Reconstruction Theorem). You need an infinite number of functions to encode all the information of the system. Equivalently, one may encode this same information in a single functional, either through a functional integral or as a wave functional (cf. 214552).
This is unchanged, except perhaps for the fact that it is usually much more convenient to evolve operators instead of states, because covariance becomes manifest. The abstract Schrödinger equation, $\frac{\mathrm d}{\mathrm dt}|\psi\rangle=-iH|\psi\rangle$ is as valid in non-relativistic QM as it is in QFT (and so is the Heisenberg equation, $\dot A=i[H,A]$). In this sense, the evolution is still unitary, but it is expressed in terms of operators instead of states.
This is unchanged.
This is unchanged, except perhaps for the fact that it is sometimes convenient to artificially enlarge the Hilbert space so as to include "negative norm states", that is, the inner product is relaxed into a sesquilinear form (which agrees with the positive-definite "true" inner product in the "true", physical Hilbert space).
This is unchanged.
Quantum field theory is quantum mechanics applied to Lorentz covariant causal systems. That is, quantum field theory is simply quantum mechanics plus special relativity. Demanding Lorentz covariance and causality constrains the systems you can talk about. For example, a crystal lattice completely breaks Lorentz symmetry, so that's out.
The systems that you can talk about turn out to be those made from Lorentz covariant local quantum fields. This is basically the message of the first 250 pages of Weinberg's The Quantum Theory of Fields. Here is the beginning of Ch.2:
The point of view of this book is that quantum field theory is the way it is because (with certain qualifications) this is the only way to reconcile quantum mechanics with special relativity. [...] First, some good news: quantum field theory is based on the same quantum mechanics that was invented by Schrödinger, Heisenberg, Pauli, Born, and others in 1925-26, and has been used ever since in atomic, molecular, nuclear, and condensed matter physics. [...] [T]his section provides only the briefest of summaries of quantum mechanics [...]
(i) Physical states are represented by rays in Hilbert space. [...]
(ii) Observables are represented by Hermitian operators. [...]
These -- in the full form in the book -- more or less cover your points 1 through 5.
I also recommend Weinberg's talk, What is quantum field theory and what did we think it is?
I think on a pedagogical level thinking of quantum field theory as different from and not just a subset of quantum mechanics may have something to do with that students are first exposed to the Schrödinger equation in the wrong form. The Shrödinger equation is, fundamentally, not a PDE in real-space. It's an ODE in Hilbert space. Correspondingly, one should not start with wavefunctions, but with statevectors, as other answers and comments have pointed out.
All of these postulates continue to hold in relativistic QFT, except that the time-evolution operator is no longer defined by the Schrodinger equation with a nonrelativistic Hamiltonian.
The only one that requires significant new elaboration in the relativistic context is the existence of an inner product. In nonabelian gauge theory, it often a useful calculational trick to formally expand your Hilbert space to a larger state space that includes negative-norm "ghosts." Such a state space is no longer a Hilbert space because its sesquisymmetric bilinear form is no longer positive definite, and is therefore no longer an inner product. But the key point is that you never have to introduce ghosts; they are merely a useful calculation trick, but do not physically exist. You can always do any calculation without invoking ghosts; See here.