The need for quantum fields
There are two seemingly unrelated conceptual steps between quantum mechanics (QM) and quantum field theory (QFT). One step reconciles QM with special relativity (SR) and the other replaces a fixed finite number of particles with fields that have infinite degrees of freedom. These two steps appear unrelated, but in order to arrive at a consistent theory both are needed.
SR postulates that all laws of physics are invariant under Lorentz transformations. QM violates this principle, because Schrödinger equation, being derived from the non-relativistic dispersion relation, is not Lorentz invariant. The solution is to derive the quantum evolution equation from the relativistic dispersion relation. This can be done in two ways with one leading to Klein-Gordon equation and the other to Dirac equation.
Now, among the consequences of the quantum theory equipped with Lorentz invariant evolution is the existence of antimatter and pair production. Therefore, it is no longer consistent to assume that the number of particles, and degrees of freedom, is fixed. The upshot is that a consistent quantum theory that incorporates SR generally looks like a QFT.
It is not true that Dirac notation is not employed in QFT. However, unlike QM, QFT does not yet have a canonical mathematical formulation, so the formalism used may differ depending on the source. That said, Dirac notation is often not the most convenient. See creation and annihilation operators for a simple and widely used alternative.
Entanglement is certainly present in quantum fields. In fact, spatially adjacent modes of a quantum field are known to be very strongly entangled, see this paper for details and a comprehensive discussion of entanglement in QFT. See also this paper for an example of a calculation of von Neumann entropy in a quantum field.