Can I use our Internal Behind the Firewall Github Installation?
Sadly no, right now you can only use Zapier with the hosted version of Github on Github.com. What you can do in certain instances is set up outbound webhooks from your Github installation to send to Zapier when certain things happen (like post-commit hooks). You can use the Zapier generic webhook service to receive those webhooks and trigger a Zap.
Why are my organization's repositories missing?
You may need to approve third party applications for your organization - follow the guide here.
Note: If you try this and don't see Zapier among the list of third-party apps that have requested access to your organization, you may need to revoke access between Zapier and GitHub, then reconnect them to select your organization properly. Go to this link to do so: https://github.com/settings/applications
Then, go back to your Zapier account, and try connecting your GitHub account again. You should then be prompted again to select the organization you'd like Zapier to have access to, and be able to approve that as needed.
Pull Requests are triggering the New Issue trigger
GitHub considers Pull Requests an issue, so they will be picked up by the New Issue trigger. You can use a filter for the PR URL to exclude these.
New mentions are not triggering the Zap when using the "New Mention" trigger
When using this trigger, you must have "Participating" notifications enabled here:
Additionally, for the trigger to work properly, you can't have viewed mention in GitHub, otherwise GitHub will mark it as read and it won't be seen when we poll.