In Japan, they often start going upwards at the same time as they are excavating. I watched as Toranomon Hills Business Tower was being built and they were installing windows on the 35th floor while dump trucks were coming out of the parking garage loaded with dirt.
Sounds crazy to go up before finishing the bottom in a country that's prone to earthquakes
They drive giant steel beams down into the ground (to bedrock?) and then attach the building’s steel frame to these anchor beams. As they excavate the basement floors, they install braces between the beams. The parking garage ramps and the subfloor are all metal welded to the braces. The excavator and dump trucks all stay on the metal ramps as they snake their way down. Other bonuses: allows for smaller dump trucks so no giant trucks on city streets and from the start of construction, no vehicle tire touches dirt.
Dampers can be and are installed during construction, although that's for bridges, but I'm sure it can also be done for buildings.
It IS done for buildings. Regularly. Even here.
Mass tuned dampers are included in pretty much any building over a certain height. They are not used just for earthquakes but also to reduce wind sway and mitigate resonance events in skyscrapers which can be incredibly dangerous.
