The nearest quasar is much closer than that. Though it depends on the definition of "quasar", many would say that 3C 273 is the nearest, with a redshift z=0.173.
Some quasars are nearly as old as the universe, yes. The highest redshift we've observed is around z=6, so at that time the universe was less than 10% of its current age.
Yes, though their numbers are greatly decreased. The peak of quasar activity was between redshifts of 2 and 3.
This is an illusion induced by a combination of relativistic and projection effects. We measure the angular velocities of quasar outflows and then use the distance to get the linear velocity. If you don't correct for emission delays due to the finite speed of light, then this can sometimes give you an outflow speed faster than c.