I'm not familiar with Littlewood-Paley theory beyond the norm-preserving equation and its significance, or the concept of "unitary" beyond "self * adjoint = identity", but here's what I do know.
The criterion assures invertibility, and the "decomposition" property of the transform (as opposed to generative), as explained in the referenced post. This comes with important benefits, including information non-duplication (the frequency axis is tiled uniformly and filters complement each other), and uniformity of representation along frequency (L1 norm, see "L2 norm" here) - which enables spatial operators along said axis like convolutions (what's done in Joint Time-Frequency Scattering).
LP & MRA
One doesn't imply the other; LP can also be satisfied with STFT. What's important is, we seek kernels localized in frequency, but also that the filterbank captures the entire input, meaning there's necessarily multiple kernels. Per LP, these kernels must also sum to unity. STFT and CWT both have a "generating rule" of next kernel from previous: under STFT, it's frequency shift, and under CWT, it's a dilation while preserving zero-mean. Turns out, the appropriate generating rule is all that's needed, and any $\psi$ will work (with caveats) - covered here.
These are all my thoughts, I don't know if it's the formal justification.
Re: question points
How the equations in Mallat's work are a "standard Littlewood-Paley condition"? I've seen Littlewood-Paley decompositions but that seems to tackle slightly different territory.
Following the comments, there's "approximate orthogonality" in that wavelets sufficiently separated in frequency are uncorrelated. But I certainly disagree with thinking of this as "orthogonality" as great redundancy is a central trait of CWT. It is important that "sufficient separation = uncorrelated", though, for sake of sparsity and time-frequency resolution.
How the equation in the first linked paper corresponds to a multi-resolution analysis?
It can certainly be misleading to write as the paper does, "MRA i.e. LP". The L2-normed CWT is a galaxy away from LP, but it's still MRA.
I know zilch about graph convolutions, but I can conceptually generalize in that, it's still spatial operators in the sense of taking a number of input points and producing a number of output points that satisfy the same properties as regular convolutions. The core difference is potential non-uniformity and non-ordinality of inputs, unsure how they deal with that.
at scale $J$
The idea is, we wish to restrict the support of the largest wavelet, since e.g. there's no infinitely large cat - and the remaining frequencies will be tiled by a lowpass filter that's designed to complement the filterbank (satisfy LP). Moreover, as wavelets are zero-mean by definition, they'll never tile the DC bin, so again a need for lowpass.
But, caveat: practically we don't immediately follow up the last CWT wavelet with lowpass, instead we tile with wavelets of same scale but spaced linearly in frequency, i.e. STFT except zero-mean enforcement. It must be that Mallat's filterbank parameter choices permit to skip this step, but it's not a good idea in practice.
why, in that same paper, equation 14 satisfies equation 15
The theoretic answer is given here. The practical answer is - quoting (emphasis mine), "The wavelet and the low-pass are designed to ..." - they certainly don't in general, we make it happen.
why one would look for equations/statements of that form
The form simply includes all filters used in the transform - completeness.