Is it possible to vertically adjust staff labels?
They are a little above the center.
You can alter the Baseline Shift for a given Staff Label in the Name Editor, but you can’t set a Baseline Shift in the Paragraph Style, oddly.
Different fonts are likely to sit at different positions, based on their bounding box.
Also, that illustration looks perfectly balanced to me. Optical centering is better than mechanical centering. (If it were all capitals (which I would not do), then I’d agree about the alignment.)
No, when Dorico draws the numbers separately from the main name, it’s using two different text items, and you can’t edit the numbers.