First of all, strictly speaking you are talking about a finite-dimensional (in fact, 2-dimensional) caricature of QFT. More precisely, the only way in which your model can be thought of as a QFT is if the underlying space-time consists only of two discrete points. For a (Minkowski) space-time continuum, the arguments of the two-point (Wightman) function $\omega$ are *not* elements of $\mathbb{C}^2$, but rather (temperate) test functions in space-time taking values in (say) a finite-dimensional complex vector space $V$ whose dual is where the fields take values in. In other words, the two-point function $\omega$ is a (two-point temperate) *distribution* on space-time.

Algebraically, the "complex symplectic" character of $\omega$ is more conveniently expressed by a property of *positive definiteness* using an involution operator on $V$ which behaves analogously to complex conjugation, just like a positive definite sesquilinear form ( = Kähler form) associated to a complex symplectic form. This property will be crucial later on to recover the Hilbert space where the field operators act, and (unlike the complex symplectic property) is spin-independent.

With those provisos out of the way, the answer to question 1.) would be *yes* - under certain circumstances. There is a class of QFT's which is *completely* determined by its two-point function: the so-called *generalized free field* theories. All you need from $\omega$ to get one of these is (besides the aforementioned *temperedness* and *positive definiteness* properties) is the so-called *spectral condition* (which restricts the support of the Fourier transform of $\omega$ - by the way, the main reason why we need temperedness is to be able to *define* that Fourier transform) and *microcausality* (which says that $\omega$ is *symmetric* or *antisymmetric* in its arguments whenever they are causally disjoint, depending on whether the field has integer of half-integer spin respectively). Neither temperedness nor the spectral condition make any sense if $\omega$ has finite-dimensional arguments, by the way. These four properties comprise a minimal subset of the so-called *Wightman axioms* for $\omega$.

Generalized free fields are thus called because they have *no* interaction, that is, their S matrix is the identity operator. This is expressed by the fact that all its truncated $n$-point functions *vanish* except for $n=2$, which is equivalent to expressing the $2n$-point functions in terms of $\omega$ pretty much as you wrote in the OP. The difference from the usual free fields is that the field operator does *not* necessarily satisfy an equation of motion coming from a Lagrangian variational principle - as such, there may be no action functional associated to it. The translation invariance of the relativistic vacuum state (entailed by the spectral condition, which on its turn *defines* such a state) implies that the one-point function is *constant*, and by redefining the field operator one can set this constant to zero - this, on its turn, forces the $(2n-1)$-point functions to vanish as well (not just the truncated ones). The ensuing $n$-point functions satisfy (appropriate versions of) the same Wightman axioms. Finally, the field operator(-valued distribution), the Hilbert space where it acts and the relativistic vacuum state vector can all be recovered from the $n$-point functions through the so-called *Wightman reconstruction theorem*.

One can think of a generalized free field's $n$-point functions as a quantum version of the moments of a *Gaussian* probability measure for a random variable, which is characterized by its *mean* ( = one-point function), which can be set to zero by appropriately shifting the corresponding random variable, and its *covariance* ( = 2-point function).

As for question 2.), the particular form of the $n$-point functions for a generalized free field and the Wightman axioms together ensure that Wick's theorem holds with essentially the same form as for free fields, keeping in mind the changes for half-integer spin due to the spin-statistics connection, which is encoded in the appropriate version of the microcausality property.