Given an arbitrary triangle mesh, is it possible to automatically generate a UV mapping for it, and if so, how is it done?
Short answer: Yes, but it won't be pretty.
Explaination: While there are algorithms to automagically UV map polygon soup, the mapping probably wont be ideal. UV mapping is an art really. Choosing where to hide the seams and where to put them to limit stretching and optimize texture space.
That said, there are lots of tools that can make manual UV mapping a heck of a lot easier. For example: http://knowledge.autodesk.com/support/maya-lt/learn-explore/caas/CloudHelp/cloudhelp/2015/ENU/MayaLT/files/Create-UVs--Automatic-Mapping-htm.html