"If this gear was a triple threaded worm gear with the same criteria as that problem, would I still figure out the pitch diameter the same way?"
I have manufactured gears and worm gears but never gone much into the math of it. I would think the number of threads has no effect in this case. Its one circumferance working with the other Circumference by whatever means. (It could be simple friction one rubbing against the other, but then we would be losing energy and quickly destroy the parts. Gears are a much more efficient way to transfer energy. The basic idea of a worm gear is to get a large reduction of rpm. The work gear can only be the drive gear, it can never be the driven gear. )