Whoa cool questions! Let's see if I can answer...

1> All time greatest achievement: Modern medicine

2> 20th Century's greatest achievement: The Internet

3> So far the greatest achievement of 21st Century: Relief efforts after Hurricane Katrina

4> All time worst achievement: The Holocaust

5> 20th Century's worst achievement: The Holocaust

6> So far the worst achievement of 21st Century: being American I'll go with... both 9/11/01 and war in Iraq