We have learned much in the last few years, but there are many blanks and deductions still.
For number 2, they used printing stones.
For number 3:
It seems very unlikely to me that multiple facilities were used to produce a single sheet. It makes much more sense that the sheet was run through both sides, cut up, and the cut cards shipped to the factories for packing. As American Lithographic, Bien, Brett and others were naming, marketing, and designing packs for the tobacco monopoly, the shipping was presumably easy as they were already sending tons of material. Multiple facilities were probably used over the course of production; the claim that all the T cards were made at American Lithographs's headquarters office in NYC is a busted myth that never had any evidence for it.
What happened to the plates? They are lost to time unless one appears. They or the production material and art required to make them must have been kept around by one of the lithography firms until 1919 at least to produce the final T206 clone set, T213-3. Nobody would have considered them to have any value at the time.
Overprintes were probably edited at the lithographer, rerunning existing stock sheets to X out the factory and overstep a different one rather than reprint the whole sheet, in order to comply with the law requiring the factory to be present on premiums. There's no evidence or reason to think that the tobacco factories did any of the printing directly themselves.
The sheets are probably larger than most imagine - other T card material strongly suggests very large sheet sizes, which surprised me. There may not have been a large number of unique subjects per sheet though; as no uncut T206 material is known outside of the Wagner strip that clearly isn't production, if that is even authentic. Miscuts have shown lots of top/bottom and some adjacent cards, but not enough to piece together anything approaching a full sheet. There is evidence that other sheets of T cards from the same AL/ATC partnership of the same physical size as T206 contained 25 unique cards (T42).
Steve B. needs to come answer the technical questions
, and Pat will know much more about the T206 specifics.