I feel your pain here. In my opinion shaft design is one of the most complex problems in structural engineer and, simultaneously, a problem that gets almost no treatment in mainstream textbooks. The problem seems to be getting exacerbated by our reliance on FEM software as well. With such "accurate" tools available to us, we seem to be losing our motivation to truly understand this stuff. Fundamentally, a core is just a cantilevered beam with a bunch of holes in it. It's astounding how complex that makes the problem however. Some excellent resources that I've drawn from are shown below. A very non-exhaustive list of things that may need to be checked includes:
1) Drift
2) Flexural strength.
3) Shear strength.
4) Torsional strength.
5) Force transfer around openings.
6) Coupling beam capacity and behavior.
7) Dynamic characteristics under wind and their influence on occupant comfort.
8) Ductile detailing in high seismic designs.
9) Potential for seismic pounding against adjacent structures.
10) Global stability in the transnational sense.
11) Global stability in the torsional sense (tall + single core)
12) Differential shortening between shaft and columns for tall buildings and, particularly, those with outriggers.
13) Shear transfer across floor level cold joints if those exist.
14) Overall axial capacity.
15) Backstay effects where basements are involved.
16) Behavior and detailing at transfer diaphragm levels where concentrated load is moving from one wall group to another.
17) Compression stability of wall ends.
18) Appropriate flange widths for both tension and compression flanges of wall groups.
19) Shear friction along the vertical joints between interconnected segments of wall groups.
20) Anchorage -- not just development -- of core vertical bars into foundation elements.
Obviously, not all of these things are germane to all building morphologies. It's a much simpler problem for modest structures.
I have all of the references below in my library. If you plan to do some shopping, I can elaborate on the strengths of each. I'll only respond to targeted questions however. I don't have time to write book reports on all this stuff for sport.