1. If a Battleship has been used in the attack air and attack sea units phases, on the phase where you can bomb airfields, can the unit still do this, or once it has been engaged in combat for a turn, does it mean it can't so shore bombardments. Or can it do shore bombardments every turn regardless if it has been already involved in combat that turn?
Each unit may be involved in combat once per step in the combat sequence that it is allowed to participate in. Since battleships have air, sea and land attack capabilities, they may fire once in each of those combat steps.
2. If after a previous round, you have ships left from both sides in the same sea zone, on the movement phase can you move out of the zone or do you have to do combat there?
You can move them out.
3. AA guns are included but not in the wording within the rules. Are these used as in previous A&A games?
They work the same way other units do. They have an air attack value of 3, so they add 3 dice during air combat step. This is different from other A&A games.
4. Can infantry attack an airfield?
No, and neither can artillery.
5. If i have a USA carrier with 2 planes, and it is attacked by 2 fighters, during the air attack phase are the USA fighters considered to be in the air so the japs can attack them?
All planes are assumed to be in the air during the Combat phase, whether carrier-based or land-based.
Then, during the sea attack phase if the japs have any planes left, can they then attack the carrier? So, can a unit be involved in 2 attacks during a turn?
Yes. See question 1.