QFT - Bra-ket notation and correct operator interpretation

I’m taking a course in Quantum Field Theory and I’m having hard times understanding the notation adopted. Let me be more precise. Considering the Lagrangian for the (classical) real scalar field [Klein-Gordon], my professor wrote that

$$ L = int d^3x mathcal{L} = frac{1}2 int d^3x (partial_mu phi partial_mu phi – m^2 phi ^2) = frac{1}2 ( langledotphi|dot phi rangle – langlephi | W|phirangle)$$

since, ignoring the boundary term after the integration by parts, we have

$$ langlephi | W|phirangle = int d^3x (nabla phi cdot nabla phi + m^2 phi^2) = int d^3x phi (-nabla^2 + m^2) phi.$$

What I understand here is that we’re just interpreting the Lagrangian as a sum of bilinear operators acting in some kind of space – let’s call it $X$ – on which we have implicitly defined an inner product

$$langlephi | psi rangle = int d^3 x phi(x)psi(x).$$

My professor also said that there’s nothing of "quantum" in this formalism, in the sense that we’ve just rewritten $L$ in a more suggestive way.
In order to find the eigenvectors of $W$ we observed that $W$ essentially acts like the operator $(hat p ^2 + m^2 )$ on a wave function (in space representation). So its eigenvectors must be the equivalent-in-$X$ of ket momentum $|krangle$ (I’m assuming $hbar = 1$). From now on, let’s denote with a subscript "X" the elements of this space (e.g. $|krangle_X$), for clarity.

Further calculations lead to the following result for the general solution of motion equation:

$$ phi(x,t) = int d^3 k mathcal{N} [a(k) e^{i(kx – omega t)} + a^*(k) e^{-i(kx – omega t)} ].$$

If we quantize this field, $a(k)$, $a^*(k)$ and $phi(x,t)$ become operators that act on a Fock space, in which we can have particle state like $|krangle = hat a^dagger_ k |0rangle$. The issue is that this ket (which – in my thoughts – is a "true ket", in the sense of QM) resembles too much $|krangle_X$; this happens with the field solution as well: I can think of it as $phi (x,t) = langle x|phi(t)rangle_X$, but I can also consider the eigenvector [another "true ket"] equation for the field operator $hat phi |phirangle = phi |phirangle$. These two couples of similar "kets" lead me to bizzarre expressions, when I get down to calculations so I need to make order in my mind.

First of all, my guess is that $X$ has nothing to do with Fock space. It must be some kind of functional space (maybe $L^2(Omega)$) of the equation solutions – but since I haven’t taken a course in differential equation I’m not sure. If I’m right $hat p$ would have nothing to do with the momentum of a particle (that would be the Noether’s charge $P^Q$) — so what’s its interpretation?
As another consequence, writing something like $hat phi |krangle$ would have no sense, since I’m considering an operator acting on a different space.

In conclusion: I have quantum operators which are expressed – classically – as integrals of $phi$ and its derivatives, so I can manipulate them with ordinary integral techniques and, as long as I define an appropriate inner product, I can interpret them as matrix elements of operators acting on some space $X$. These matrix elements, when I substitute the operator expression of $phi$ in the integrand, become true quantum operators acting on the Fock space.

Can anyone tell me whether this interpretation is correct or not (and eventually show me the connection between $X$ and the Fock space)? I’m afraid I could have interpreted it all the way wrong, since I haven’t found any digression about the notation in any book.