Ok, one more question:
At the beginning of the game, Japan has at least 1 unit on each island. In the case of Choiseul or Santa Isabel, if you remove that single infantry, do you lose control? Do the reinforcement points suddenly go to the natives? Or is it still yours until someone else lands a unit on it, hence the Japanese flag icon indicating you have control in the box with the single unit?
If the US player has no ground units on that islands, than you keep control.
Second part:
If you have 1 AAA on and island, since it has no land attack value, can it keep control of an island for you?
If the other player has no units on that island: Yes.
If the other player has land units on that island, but your navy near that island has in summary more supporting points than your enemy with ground units and navy: Yes.
In short: You only lose control of an island if the enemy has ground units on it and the dice value of enemy force is bigger than yours. Naval units only count if there is at least one ground unit (can be an AA gun) on island. Air units not count in.