Relation between spinors and anticommutation relation of fermions

Looking at the expression of a quantized Dirac field $$ psi = sum_s int a_s(p) u_s(p) e^{-ipx} + b_s^{dagger}(p) v_s(p) e^{ipx} frac{d^3 p}{(2pi)^2sqrt{2E}} , $$ one finds that it contains annihilation and creation operators $a_s(p)$ and $b_s^{dagger}(p)$, as well as spinors $u_s(p)$ and $v_s(p)$. The anticommuting properties of the field is provided by the annihilation and creation operators, where is the solution of the Dirac equaltions are provided by the spinors. As a result, the spinors on their own won't give you the anticommuting properties.

The first paragraph as people have pointed out in the comments, is indeed correct.

About the second one, the short answer is no you can't do the same, but more over the proper statement is you don't need to do the same. Within the framework of field theory, in contrast to classical quantum mechanics, multi-particle states are already taken into account. That is the construction of the Fock space (see Second quantization) is done by acting with creation operators on the vacuum. Acting with creation operators labeled by different momenta or coming from different fields, leads to states that represent multi-particle states, so there is no need to take tensor products of spinors.

Nonetheless, fermionic fields (solutions to Dirac's equation) are subject to canonical anti-commutation relations (as opposed to the bosonic case where these become commutation relations): $${psi_a(vec{x}),psi^dagger_b(vec{y}) }big|_{t_x=t_y} = delta(vec{x}-vec{y})$$ which induce the correct commutation relations on the creation and annihilation operators $a,a^dagger$ of the usual mode decomposition.

So the difference of both approaches in the end is what the fundamental object of study is under each description. In quantum mechanics wave-functions describe a single particle, in quantum field theory, fields describe the collective interactions of particles.

More over, in quantum field theory one must also respect Lorentz symmetry among other possible symmetries of a theory. So any interesting object must be a Lorentz invariant object. Additionally one imposes also locality so for operators within the Lagrangian of your theory, there is the requirement of depending on a single space-time point.

One could consider a two point function, or correlation function, this one is actually of interest within QFT: $$langle psi^dagger(x) psi(y) rangle$$ Which is just the propagator of the field $psi$ in a free theory and tells you how the field evolves between the space-time points $x$ and $y$ in the absence of interactions.