I’ll begin with the design of a modest-sized unit that is about half the size of the often-recommended 20-ton elevator carriage, which is imagined to be able to carry a 17-ton payload up in only a week.  (Which I believe impossible)

First, in order to ascertain the starting speed to drive the elevator all the way [...]

The space elevator is absolutely dependent on a strong tether cable, currently aimed at 100,000 km long, but it could be shortened considerably with a larger mass as the counter-weight. Carbon nanotubes are meant to be the way forward, providing the huge strengths needed for such a cable, but this is problematic and it will [...]