Anharmonic field theory: pedagogical simplifications

Thank you all for so much help as I work through Zee’s QFT book. Here I finally have a question about physics instead of math.

Zee makes several comments that the main thing left to do in QFT is find a way to solve

$$ Z(J)=int!Dvarphi,exp!left[ iint!d^4x,frac{1}{2}big[ (partialvarphi)^2-m^2varphi^2 big] -frac{lambda}{4!}varphi^4+Jvarphi right] .$$

Since we cannot solve it directly, even by the magical process for "discretizing and undiscretizing" infinite dimensional path integrals, Zee makes a study of two simplifed cases. He presents a "baby problem"

$$ Z_B(J)=int_{-infty}^infty!dq,exp!left[ -frac{1}{2}m^2q^2-frac{lambda}{4!}q^4+Jq right], $$

and a "child problem"

begin{align*}
Z_C(J)&=int_{-infty}^inftyint_{-infty}^inftydotsint_{-infty}^infty!dq_1,dq_2,dots dq_N,exp!left[ -frac{1}{2},qcdot Acdot q-frac{lambda}{4!}q^4+Jcdot q right]
&=iintdotsint!dq_1,dq_2,dots dq_N,exp!left[ -frac{1}{2}sum_{m,n=1}^N q_m A_{mn} q_n-frac{lambda}{4!}sum_{k=1}^N q_k^4+sum_{l=1}^N J_l q_l right].
end{align*}

Analytically, I see the difference is that $J$ and $varphi$ are functions of a continuous variable in $Z$, a discrete variable in $Z_C$, and they are not functions of anything in $Z_B$.

Q1: What would be the physical significances of $Z_B$ and $Z_C$ respectively?

Q2: What does it mean for $q$ not to be a function of anything?

For $q$ to be a function of a discrete variable, I see that $q$ describes, essentially, the vertical displacement position of a set of oscillators located at some number of discrete $vec x_k$. If it’s not a function of anything, does that mean it is a single oscillator? How could it not be a function of time at least? Is the location of the single oscillator coded into $q$ itself so $q$ means "the oscillator at some $vec x_0$?" If you can go into detail about the physics we are looking at here, that would be very helpful.