The following issues were found:
- Global proxy settings (Menu / Control Center / Network Proxy) are not honored by either the package manager, Firefox, or command line tools. (Actually I did not find any application that does honor these settings, and yes, I did reboot after setting them.)
- Environment variables set in /etc/bash.bashrc or ~/.bashrc are not honored by Firefox either, even when the latter is started from the command line.
- The Package Manager doesn't recognize system proxy settings, but you can set a proxy manually in the preferences. However, this does not affect e.g. command-line apt-get.
So I ended up with "system" proxy settings in the control center (which are apparently completely ignored), in Firefox, in the Package Manager, and as environment variables. Setting env_keep="http_proxy https_proxy ftp_proxy" in /etc/sudoers at least made sudo apt-get work.
But then I opened Control Center / Language Support, was told that there were missing language packs (as was expected)... and found that this dialog doesn't honor any of the aforementioned settings. It simply hangs trying to download the language packs. Since there is no way to enter proxy settings anywhere in the dialog, my attempts ended there. (No, I did not feel like copying the list of language packages into a terminal for manual apt-get'ing them. I gave up at that point.)
Perhaps this post can be put in the appropriate channels by people more involved with the Mint community: Currently proxy handling is a disgrace. I admit that some of the issues aren't strictly in the domain of the Mint maintainers, but I don't really care about the maintainer side of things. From a user perspective, it sucks, and royally so.