Are accelerated spacetimes in General Relativity "vacuum solutions"? Why are they not integrable as the Kerr-NUT-dS? After all, the following cube of metrics seems to mean that every Kerr-type metric can be derived from the accelerated versions,...Why are accelerated solutions "hard" to handle with? Any physical meaning of them, specifically the C-metric? I mean, I did know that Kerr metrics are rotating black holes, or the NUT parameter is some type of gravitomagnetic mass term, but what is the physical picture of, e.g., the C-metric (and more generally accelerated spacetimes)?
Please, have a look at a paper
Griffiths, J. B., Krtouš, P., & Podolský, J. (2006). Interpreting the C-metric. Classical and Quantum Gravity, 23(23), 6745, arXiv,
which should answer all your questions about the nature of accelerating parameter, with lots of pictures describing 3D casual structures of accelerated space-times extended across horizons.
One thing to note, is that this type of solutions exhibits conical singularity along (half)axis (in a given patch of a space-time), which could be interpreted as semi-infinite cosmic string pulling the Schwarzschild black hole and thus providing the source of acceleration. This also answers your question on whether this solution is vacuum solution: yes, it is, except for the cosmic string which correspond to $\delta$-like singularity of Ricci tensor.
As for the 'hardness' to work with, why shouldn't they be? You are breaking the symmetry of Schwarzschild solution by accelerating it and in a completely different way than the rotation does, so disappearing integrability is not unexpected.