My direct experience with BEAST is limited, but I think I can partially answer your questions. With the 3rd population, a very short coalescence time is exactly what you would infer if there were only a single haplotype among so many sequences. Accumulation of mutations over time is what gives rise to different haplotypes, and is about the only thing that can do so in (supposedly) non-recombining sequences like mtDNA genes. Thus, the longer the time since the MRCA, the more haplotypes one would expect.

While the absolute TMRCA ranges might or might not be reliable in the BEAST output, the relative ranges in comparison with each other seem realistic given the sample sizes and numbers of haplotypes. BEAST is highly rich in parameters, with a lot of possible models that make a lot of assumptions, which gives me concern about putting too much reliance on the accuracy of the output, but in a relative sense what you report makes sense.

For the third population all 23 sequences are same, so they will coalesce contemporarily, at 0 generation backward in time. At least two haplotypes are need to observe a coalescent event so that you can have an estimate for TMRCA.

The coalescent time to MRCA for one polymorphism is necessarily much shorter than for 4 or especially 10 polymorphisms. That is simply coalescent theory - maybe read up on it.

It is because the probability of coalescence for one change is much higher than the joint probability of 10 changes.