1) Why are odd numbers of protons or neutrons in cluster decay so rare?
Experimentally, even-even nuclei (i.e, a nuclei with even $Z$ and even $A-Z$) are more stable than even-odd or odd-even ones, and both configurations are more stable than the odd-odd one. A stronger p-p (proton-proton) and n-n (neutron-neutron) interaction with regards to p-n interaction inside the nuclei accounts for it.
This is explained in the shell model or in the Fermi gas model by the Pauli exclusion principle. In these models particles are assumed to be all stored below the Fermi level, and effectively free (in a potential well that binds them in the nucleus) since almost any interaction would move them to already occupied energy levels; the only allowed interactions are those that exchange the particles' energies. This strongly discourages p-n interactions, since p and n quantum numbers are different and in general also an energy exchange would violate Pauli's principle. Pauli's principle also guarantees that nuclei with equal Fermi level for both p and n are more stable (this can be seen by a straightforward calculation).
2) Why is the number of neutrons always an equal or greater number of than protons in the cluster?
The potential well that binds neutrons and protons in the nucleus is deeper for neutrons than for protons, due to E.M. repulsion between the latter. Since Fermi levels must be the same, configurations with more neutrons than protons are favored.
I don't have a definitive answer for the last two questions, but I'll try to formulate some hypotheses anyway.
3) Why couldn't a nucleus emit a single proton?
I think this could be explained computing the $Q$-value of nuclear decays and observing that is always smaller for proton emission than for cluster (or $\alpha$) emission, making the former far more difficult to be observed. Note however that fission processes do in general involve protons as side products.
4) Why doesn't cluster decay involve odd numbers of both protons and neutrons in any of the parent nucleus, the emitted cluster, or the daughter nucleus?
The answer for the emitted cluster and the daughter nucleus resides in what is said above. It seems therefore that for an even-even + even-even decay also the parent nucleus should be even-even, preferring to decay in $\beta$ otherwise.