What is the meaning of spin-splitting energy as described here?
I see now how your initial questions were related, as they all fall under the scope of crystal field theory. I wrote a bit about this in a previous answer.
At least in the context of molecular crystal field theory, you will usually see the phrase pairing energy rather than "spin-splitting". The distinction is basically just the direction, where the pairing energy is the penalty for having electrons share the same spatial orbital and the spin-splitting energy would be the increased stability from separating a pair.
I don't believe the spin-splitting or field splitting can be read from the DOS. The way they are mentioned in the paper seems to suggest they are giving a qualitative explanation for seeing all the compounds in high spin configurations. The spectrochemical series gives an ordering of ligands based on how much field splitting they should cause. Since S is known to be a weak-field ligand, it makes sense that the spin splitting would energy would be larger than the field splitting, leading to high spin configurations.
The word "splitting" seems to occur only in one sentence, over the course of the entire paper that you referenced:
"The d-orbital spin-splitting energy is stronger than the weak crystal field splitting energy of the S ligand around the TM atoms"
I can see why you had to ask this question here, because when I search "d-orbital spin-splitting" in quotes (so that we're searching for that exact phrase), only two results come up: this paper and this paper (the same happens if I replace either of the hyphens with spaces.
In this particular paper, it seems that what they're referring to when they say "d-orbital spin-splitting energy" is the difference in energy between the high-spin and low-spin configurations. The paper goes on to say that because the d-orbital spin-splitting energy is stronger (i.e. larger) than the weak (i.e. small) crystal field splitting energy ($\Delta_0$), the system is said to be in the "high-spin state". This would make sense since high-spin states are sometimes characterized as having a small $\Delta_0$, for example, in the Wikipedia page in the last link I provided, it shows "$\Delta_0$ small" for the high-spin states:
and "$\Delta_0$ large" for the low-spin states:
My reading of the relevant paragraph suggests to me that the authors made this conclusion about the "d-orbital spin-splitting" being larger than the crystal field splitting of $\Delta_0$ based on Table II, rather than any of the DOS figures. So perhaps the readers are not expected to deduce this by looking at the DOS plots as you ask how to do, but it could be possible, and that's why this nice Q/A system allows us to get answers from multiple people 😊.