Particle creation by a source as explained in A. Zee's QFT in a Nutshell
First, think of a source to be like a radio antenna (it is, after all, a source for electromagnetic fields). An antenna that can emit well can also absorb well. So $J(x)$ can model both a source and a sink, each with some spatial profile. In Zee's setup, that combination of source and sink is modeled as $J(x)$ with an extended spatial profile.
When Zee splits $J \equiv J_A + J_B$ you can think of them as two antennas A and B. The cross terms such as $J_A^* J_B$ term corresponds to A emitting and B absorbing and the self/diagonal terms like $J_A^* J_A$ correspond to an antenna interacting with it's own radiation pattern. If you can self-consistently account for the self-interaction in determining your antenna's radiation pattern, then the physics of communication is essentially in the cross terms.
The first expression you've written explains that $\mathcal{W}[J]$ is a sum over all possible ways in which a particle is emitted at one point and is absorbed at the other point---that is why the source and sink are "contracted with" the propagator, in the language of Feynman diagrams. Now you can understand why this measures the rate of the particle creation-transmittion-absorption process.
It is crucial that the source and sink needn't have significant spatial overlap to transmit waves/particles/fields (otherwise we couldn't use EM fields to communicate over long distances!). However, your sender and receiver better share the same frequencies! Otherwise the receiver (sink) will absorb only a very small fraction of the signal power sent out by the transmitter (source). That is why we would like $J_1(k)$ and $J_2(k)$ to have a significant overlap in frequency space. Further, you need to not just have a significant overlap between the source and the sink spectra, but your medium should non-dissipatively transmit the relevant frequencies! The resonance in the field modes $\phi(k)$ is where the propagation happens best (in the medium of the field background), with least dissipation. In field theory, this is called an "on (mass) shell particle" and basically corresponds to the EM waves we're used to dealing with.
In summary, basically, if you wish to transmit and receive a particular frequency effectively, then you'd like your sender and receiver to resonate at a frequency that the medium supports well.
I can answer your first two questions, which admittedly are not the most important ones, but maybe this will help for now.
Overlap just means that both functions are nonzero in a similar region of $k$-space. Suppose that wherever $J_1(k)$ is nonzero $J_2(k)$ is zero and vice versa; then the integral would be zero. On the other hand, if in some region both functions have significant amplitude, their product will integrate to some moderately high number.
The resonance spike is a property of the function $1/(k^2-m^2)$, it has nothing to do with the overlap per se. However, let's go back to answer 1: suppose that $J_1(k)$ and $J_2(k)$ overlap in some region in which $k^2-m^2$ is very big. In this case the integral will not be very big because $1/(k^2-m^2)$ is small. If you want a big integral you need the three functions ($J_1(k)$, $J_2(k)$ and $1/(k^2-m^2)$) to be big in the same region.
I don't really know how to answer 3 and 4. Indeed, I share your confusion, since after all $W$ goes in an exponential, so we only care about its value modulo $2\pi$. I would say that if you're reading Zee, get used to this. Not everything is proved rigorously; often he just talks about the intuitive picture without actually showing that the math corresponds to that. My advice is to just go on reading without getting too hung up on the details, and then read a more detailed book and come back to Zee later; things will make more sense.
To answer the last two questions, it is easier not to think in terms of $W[J]$ directly, but in terms of its first derivative $$ \frac{\delta W}{\delta J(x)}=\langle \phi(x)\rangle = -\int_y D(x-y)J(y). $$ Using the inverse operator of $D(x)$, $\square^2+m^2$, one gets $$ (\square^2+m^2)\langle \phi(x)\rangle = J(x), $$ which looks like a wave equation for the (average of the) field, in presence of a source term $J(x)$. Compare for instance to the Maxwell equation of the vector potential in presence of a source.
One the other hand, $W[J]=\int_{x,y}\langle \phi(y)\rangle(\square^2+m^2)\langle \phi(x)\rangle$ is related to the energy of the system in presence of the source, which will be large only if the sources are close enough (of the order of $m^{-1}$), which is the "resonance" condition.