Building cores are pretty big, and they don't earn any rent money, so they try to keep them as small as possible now.
They stack elevator shafts in the super-tall skyscrapers to help reduce the number of shafts that they need. (hehehe....shafts...) Double-Decker elevator cabs and sky lobbies help with it too.
They do tend to use concrete more often on the tall ones now because it's stiffer (hehehe....) and fireproof.
The inertial dampers from the video above help control the sway. Building codes mandate that the sway has to be below a certain amount to keep people from really feeling it. (If you go up to the observation deck of the Sears Tower, and go into the bathroom, you can see the water in the toilet move a little on windy days. It's really hard to feel the building sway.)
The negative wind pressure and turbulence on the leeward side is pretty much controlled by wind-tunnel testing and computer models.
The John Hancock Tower in Boston was notorious for having wind problems. Windows popped off of the thing for several years, and it swayed too much and needed to have a crap-load of additional steel reinforcing welded to the frame after it had been built.
The Citi-Corp tower in New York had issues too. It didn't sway or have glass plummet to the ground, but after it had been built, one of the engineers was talking to a student who had asked him some questions that got him thinking about his calculations. He discovered that if a heavy sustained wind came from a direction that they didn't quite expect, that the thing could fall over. As fate would have it, a hurricane was going to hit New York from that direction. They (sort of) secretly went in at night and reinforced the frame in the nick of time.
Tall buildings are pretty crazy. 10 stories is the tallest that I've worked on, but a buddy of mine worked on the Shanghai World Financial Center. He did a lot of the curtain wall detailing and figured out how to make the window washing system would work. They were trying to use robots that ran in tracks up and down the side of the building, but they went old-school instead.