Data storing: Where does the Test Data Orchestrator stores the data?

The Test Data Orchestrator stores data only locally in your Jira Data Center database and nowhere else. We also do not have any access to the test data stored in any Test Data Orchestrator installation. Your data is safe with you!

URL - Why does clicking a TDO-URL within a Jira-Issue does open within the same tab?

This is because Jira always opens links within the same tab, as long as you not force it to open in a new tab (right-click > open link in new tab)