Can I watch terrestrial/over-the-air broadcasting with Apple TV?

Not directly. The Apple TV box takes its input from the home network connection.

Indirectly, it is possible to install an antenna feed tuner box (e.g. HDHomeRun) that re-distributes that signal onto the home network, for the Apple TV box to pick up on, through a dedicated app (Channels).

No

in my country most content providers that offer over the air material also have an app for appletv where one can view the same content both live and their back catalog