Upgrading from 4.7.18 to 4.7.19 breaks soap
It turns out it was civicrm file permissions (in sites/all) conflicting with my server's PHP access settings.

My server dictates that if any part of a path is group writable, php file access will be denied. So I (well, my hired support person because I got desperate) went and set the directories involved to permission 755, and the file to permission 644 (civicrm/extern/soap.php).

My secondary problems with cron (civicrm/bin/cron.php) and not being able to browse for images in civimail (civicrm/packages/kcfinder/*) were fixed by following the same logic.

I'm still not sure why this was triggered by an upgrade from 4.7.18 to 4.7.19. I must have made a misstep during file copying.