1.) How do we know the dimension of the Hilbert space we are working in? Is it the number of dimensions of the $\psi(x)$?

For $N$ dimensional Hilbert space $\psi$ will be a vector of dimension $N$ and when $\psi(x)$ is a function (i.e. a infinite-dimensional vector) than the dimension of Hilbert space will be also infinite?

2.) When we have $[A,H]=0$ than that operator $A$ commutes with the Hamiltonian and the quantity is conserved and also we can measure one without distrurbing the measurment of the other. Cool.

I found out of examples: $[x,p_y]=0$
$[L^2,H]=0$ - for Hydrogen atom (spherically symetric)

Do you know of any more examples (maybe less known) ?

I think I got it. The effective action is a functional of $\phi_c$, $\Gamma[\phi_c]$. The classical field $\overline{\phi}$, i.e. the one that reduces to $v$ in the zero current limit, is only the solution to the EoM,
$\frac{\delta\Gamma}{\delta \phi_c}\bigg\rvert_{\phi_0}=-J(x)$. Now we're working with fixed current (i.e. $\phi_0$ is a function of $J$), so sending it to zero we get the desired result. $\phi_c$ is just an arbitrary variable

Hm. I dont know if they are related.
Physics just models reality via approximations so it can not give any absolute answer (what ever that means).

And ontology asks questions that I think will never be answered xD.