Dirac notation
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
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.
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