Matt,
Annoyingly (very), I had already typed an answer to your post, only to lose it because I was logged out while composing it.
Second time lucky!
I can't answer the question with 100% certainty, but here is my understanding. Mostly that is because I have never found a definition of "Documentarily Qualified".
The Consulates report numbers to DOS each month in advance of the VB Cut Off Dates being set. Therefore the Consulates must know about the applicant for them to be included in the DD.
I believe the number forms part of the Demand Data at the time (or shortly after) the
Appointment Package for Immigrant Visa Applicants (formerly Packet 4) is sent to the applicant.
This includes the Interview Date and Instructions on actions to be completed prior to attending the Interview such as the Medical Examination.
There is an earlier contact by NVC with the
Instructions Package for Immigrant Visa Applicants (formerly Packet 3) where the intent to proceed is expressed and various biographical details, Affidavit Of Support form and Processing Fee are requested. I don't believe the applicant is included in the Demand data at this point.
Details of the renaming from Packets can be found in this
DOS telegram.
Details of the Consular Processing process can be found
here.
It seems to take about 2 months from becoming Current to the Interview Date judging by this
Sample Interview Letter and comments that Ron Gotcher has made previously. In truth, I don't know for sure.
I hope that helps. I only profess to have a fairly basic understanding of the CP process.
I haven't had time to look into and respond to your second question.