Trigger a rule when a module update is available It would be terrific to have Rules integration for when the Update Manager module finds an update. Is there a way to do this?

Potential ways I'd use this would be to have an update page, customized emails, adding logs in Heartbeat, and so on.
I know that you specifically asked about Rules, but you might also consider using Drush for this.  For example, if you want an email to be sent to you every time an error appears on your status page, you could just drop this one line into your crontab:

0   */1   *   *   *   /usr/bin/env PATH=/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin /bin/drush --root=/path/to/drupalroot --uri=mysite.com status-report --severity=2


(n.b. your path may vary.)  If there is output from the status-report command, cron will send it out in an email; if there is no output, nothing happens.

Hope you don't find this answer to be too off-topic.  Rules is great for responding to user actions, but for scripting site maintenance tasks, Drush is often more convenient.