Sure, you can re-arrange the information in the template. Please, go to the Box Template step of the configuration wizard, switch to the HTML mode tab and find the corresponding blocks:
Is it possible to have the org-chart display birthday/hire-date? These 2 days will be added in the ExtensionAttribute10/11 for the user in active directory.
If you want to display information from a field that doesn't have a managed search property please add such property, update information in the field to trigger the crawling and wait until information will start appearing in search.