Changing proxy settings and making them effective
Summary answer to your questions:

What operations exactly do drush dl, pm-update, l10n-update, and the UI equivalents use? Are there differences that might cause one to work and others not?

**
drush dl: donwload a module or theme
pm-update: update Core
l10n-update: update translation
**

Do both the D7 and the D8 formats in settings.php work?

No there are Some differences between D7 and D8

What is the difference between using settings.php and the variables in the database?

In Drupal 7 there is not any different. The settings store in DB. But in Drupal 8 None on Setting Configuration store in Database So there is no way to change them via DB

How do the database variables get updated? Does the fact of changing them in settings.php also update the database?

Yes, When you changed a config in settings.php  the equivalence variable in database got updated but in Drupal 8 There is no config or variable in DB to store settins.pp configuration.

you can get settings in Drupal 8 by Settings::get(YOUR_CONFING_NAME).