Definitions and rules change from edition to edition. In the rules 'turn' means players turn (page 9). The space marine codex says a librarian may use a psychic power each 'player turn' (as an example). Same goes for every other codex. The thing then boils down to this:
1) Psychic Shooting attacks may only be done in your player turn so those are out the window.
2) Force Dome is used in the librarian's movement phase so that is out the window.
3) Quickening is used at the beginning of the Librarians assault phase so that is out the window.
4) Null Zone is used at the start of the Librarian's shooting phase so that is out the window.
5) Might of the Ancients is used at the beginning of the Librarian's assault phase so that is out the window
6) Gate of Infinity is used in the beginning librarian's movement phase.
That means that all Codex: Space Marine psychic powers can only be used in their player turn........but only because the powers specifically restrict them to doing so.
Some powers in other armies have no such restrictions (blood angels for example).