by Rex Lorenzo.
Following on the LTI idea, you can use this plugin LTI provider tool: https://moodle.org/plugins/view.php?plugin=local_ltiprovider
This will allow you to make Moodle an LTI provider, meaning that suppose you want to just use the Quiz tool for Moodle in your other system.
You can use this plugin and it will handle the single sign-on and grade transfers.
This also should work for any of the other "core" tools that Moodle provides.