Are all LEGO parts made in a single factory, or are there multiple factories? Where abouts is it/are they? If there are several, are there different factories for different pieces/sets, or do all factories produce all parts?
LEGO moulds bricks in the following plants: Billund in Denmark, Nyíregyháza in Hungary and Monterrey in Mexico.
In addition, it has another facility in Kladno, Czech Republic, that handles painting and packaging for some of the bricks produced in Denmark and Hungary.
The headquarters in Denmark is still where the most of R&D and product development happens.
The National Geographic Channel has a series called Ultimate Factories. As it happens, the episode that aired tonight was all about the LEGO factory in Billund. The show also looks at the LEGO manufacturing process, product development, and distribution network. If you want to know more about where and how LEGO parts are made, you couldn't do much better than to watch this episode. I'm sure they'll show it again sometime soon -- check you local TV listings.