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Wood thickness for box dulcimer?
There really is no "best". Are you going to build a large-ish Tennessee Music Box type instrument -- say 3-4" deep, 12-14" wide and 26-28" long? Or are you thinking of a more common size, say 2" deep, 6-8" wide and 28-30" long?
The TMB type dulcimers were traditionally built with 3/8" thick boards all around, and a solid 2x2 for the fretboard. More common sized box dulcimers can be built from 1/4", 1/8" or 1/10" thin stock. You can get nice 1/4" poplar from Home Despot or Lowest in their "craft wood" department. There are several good sources for 1/8" thin lumber that we can point you to. I'd say most builders of common size instruments use 1/8" thin lumber.
I'm currently building a museum replica of a mid-19th century Pennsylvania Zither -- a pre-dulcimer ancestor, as well as a box for one of my student's cardboard starter dulcimer.
Unless you're dead set on having fancy geared tuners, I would recommend autoharp tuning pins -- they're cheap. easy to install, easy to tune, and hold a tuning well.