Why would enabling a new module result in the majority of modules disabling?
It appears you're being affected by a known issue that has been resolved in Drupal 8 but still needs to be back ported to Drupal 7. The PHP extension Suhosin causes modules with parentheses in the package name to be disabled. You can edit your Suhosin configuration to remove parentheses from the array index blacklist or, if you're unable to do that in a pinch, you can edit the relevant module files to remove the parentheses from their package names.

More info in the core issue: https://www.drupal.org/node/2665152