Does it matter? Should I use one or the other? Or both, even? Can the license file be any name? (With any extension?)
I'm sure there's a "best practices" guide that contains all this, but I'm not having any luck finding it.
COPYING is standard in the GNU project. I think that either of COPYING or LICENSE would be understood. The FSF has a guide on how to use the GPL, which suggests COPYING, while the Producing Open Source Software book suggests either COPYING or LICENSE. The Apache License recommends a boilerplate that just points to the license online, though the Apache project seems to also include a LICENSE file in their code containing the relevant license or licenses.
It's up to you whether you want to add the .txt suffix; on Windows and Mac OS X it tends to be helpful for default file associations, while on Linux or most other Unix-type operating systems suffixes are optional and it's generally assumed that files are plain text.