1. Does the WTS still refer to the mass of JWs as the "Other Sheep" and "Great Crowd"? (Or do they use "Great Multitude"?)
It's more accurate to use "other sheep". The great crowd refers to those other sheep who will survive Armageddon.
2.
By what other Bible-based designations does it identify the earth-bound
class? (I suspect that terms such as Publisher, Overseer, and Pioneer
are not Bible based, but I could be wrong. It has been known to happen.)
Usually they use "other sheep" when referring to present day, and "great crowd" when talking in an eschatological context.
3. I see the term "spirit-anointed". Where do they get this? Is it from the gospels or from Pentecost?
Neither. This term was concocted by Watchtower. In my opinion, referred to the 144,000 as 'anointed' Christians stems from the fact that they will rule as kings and priests during the Millennial reign of Christ. 'Spirit-anointed' might be a reference to Romans 8:14-17 where the holy spirit bears witness of someone's being called for heavenly life.
4. What are the names it currently uses to identify the 144,000?
Anointed Christians or "little flock"
5. Does it openly refer to the Order of Special JWs, apart from that brief entry in the Year Book statistical summary?
The Religious Order of Special Full-Time Servants of Jehovah's Witnesses includes (as far as I know) Bethelites and traveling overseers and their wifes. They refer to the Order in the contract you need sign when accepted to serve at Bethel,
6. Do they still see members of the 144,000 as having been gathered in every age since the start of Christianity?
Yes. Nothing has been published to change this viewpoint.