Do all regular Duplo bricks have the word "Duplo" written on their studs? If not, what variations are there?
There really isn't a "on" for the Duplo studs. All Duplo studs are hollow studs. In the interior of the stud, depending on set and the year of the piece, the stud will read:
Before 2002 LEGO used "Duplo" inside studs. In 2002 they tried to drop the Duplo brand and possibly for this reason all new Duplo moulds have "LEGO" in the studs. Rare pieces like the 2x10 brick still use an older mould, so still have "Duplo" in the stud. Also from 2012 some sets at least in Europe have 2x4 bricks with "Duplo" in the studs. It's possible these are repaired old molds.