Quantum field theories are a subset of quantum mechanical theories. So they obey all postulates of quantum mechanics, they have Hilbert space, linear Hermitian operators i.e. observables, obey the superposition principles, calculate probabilities from squared absolute values of complex amplitudes, and so on.
The Fock space is a particular example of the Hilbert space – the Hilbert space for a higher-dimensional or infinite-dimensional harmonic oscillator – and it (along with the relevant free Hamiltonian) describes free quantum field theories (those with the quadratic Hamiltonian/action, i.e. no interactions).
The interacting quantum field theories don't have a Hilbert space that is "exactly" equal to a Fock space but the Fock space is still a good tool to study physics of interacting quantum field theories approximately, by "perturbative expansions".
Similarly, the number operator is only "truly well-defined" for free quantum field theories. The value of the number operator for a general state in an interacting quantum field theory isn't really well-defined. For example, an electron in the interacting Quantum Electrodynamics is "decorated" by lots of photons and electron-positron pairs appearing in its vicinity. Their number is nonzero, in some sense infinite, and it isn't even sharply defined because the number operator no longer commutes with the Hamiltonian in interacting QFTs.
No, number operator eigenstates are not the "only basis" of the Hilbert space. Quite generally, every Hilbert space (and every higher-dimensional linear space!) has infinitely many (and it is a huge infinity) possible bases. For example, eigenstates of any other Hermitian operators may be turned into a basis.
The position basis and momentum basis (or representation) are particular bases (or representations) for non-relativistic quantum mechanics. They're composed of the continuous eigenstates of the operators $x_i$ and $p_i$, respectively. But these are not well-defined operators in quantum field theory – after all, even the number of particles is variable in QFTs so there can't be any "fixed number of particles' positions or momenta". Quantum field theories have other operators (observables). The existence of the "momentum basis" or "position basis" is a particular property of a class of (non-relativistic) models of quantum mechanics; this existence is not belonging among the general postulates of quantum mechanics and these theories (with a fixed number of particles with positions or momenta) don't describe our Universe accurately.
