I have Austrian citizenship, can I live and have residency in the Turks and Caicos Islands?
The Turks and Caicos Islands are one of the “overseas countries and territories” of the European Union. As such, and in spite of their status as a British Overseas Territory, they are not part of the internal market and the EU freedom of movement does not automatically apply there.
Beyond that, they presumably have some process to become a resident, of which I know nothing. But the local government can apply restrictions as they see fit and you can't rely on your freedom of movement to obtain the right to reside in the territory.