Yes, Caroline Islands fighters would have to land on a carrier. However, two of the fighters that took off from the carriers could land in the Caroline Islands due to the airbase there. making room for them.
Taking off from or landing on a carrier doesn't count as a movement, because the fighter is remaining in the same space. Only crossing from one space to another (land or sea) counts as a movement.
The movement numbers for air units (4 for fighters, 6 for bombers) apply to the combination of combat and noncombat movement. If a figher moves 3 spaces in combat movement, it has only 1 move left for noncombat movement.
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