We interpret it as you seem to be interpreting it.
They are placed after all movement is done (during the place production phase) so the air unit limitation that are in place are the ones that are present at the time of the end of movement. But since they are in place during the other player's turn, then yes they are a valid place for the rebasing of air units disturbed during the opponents attacks.
And no, you can't place one on a territory that you just took this turn. Which causes a bit of a lag in the placing of an airbase.
Here are the two appropriate statements that are in the rule book:
During the production phase, a player may spend 5 production points to construct an airbase at any territory they own at the beginning of the turn. At the conclusion of the turn, during “place new units on the board” phase, a 3 plane capacity airbase may be placed in the territory.
Territories with an airbase can accommodate up to 3 planes instead of just one during any phase after the airbase has been placed on the board.
Hope that helps.