That territory is considered full-friendly after you captured it during your turn.
So now that its after your turn, its defiantly friendly to the point where you can land planes there.
tl:dr Yes
I think you meant to say “definitely”
The rules seem to imply you can build a new airbase or naval base on an island without having any troops stationed there, as long as you owned territory at start of turn.
Is this true?
Yes, as long as it is a named island. Island’s without names are not playable spaces in the game and cannot be used for anything, even landing your own planes, troops etc.