Welcome to Axis and Allies, and welcome to these fora. You will find a wealth of experience and knowledge here.
Since you posted in the Axis and Allies Pacific 1940 forum, I am assuming you are using this game. There are a number of other games in the Axis and Allies series and the answers to your questions could vary depending on this. For example, Axis and Allies Guadalcanal is quite different from Axis and Allies 1942.
But assuming Pacific 1940 …
Fighters may participate in any kind of combat except attacking enemy subs without the benefit of an allied destroyer being in the same sea zone (flight range permitting). Fighters would attack land troop or ship equally well. However, if you were to attack a sea zone with an enemy fleet and an enemy controlled island, they could only participate in one of these two battles in a single turn.
Damaged battleships are still as tough (att/def = 4/4) whether they have been damaged or not.
Yes, combat between ships and aircraft is just like any other normal battle except there is some requirement to make sure the planes could have landed safely. You are not allowed to send planes on deliberate suicide missions.
The answer to the fourth question is no. You must be conductiong an amphibious assault to use a bombard attack.
There are many things to consider as Brittan, but my number one bit of advice is to buy plenty of infantry to protect your starting planes and keep Brittan alive as long as possible. You may have to give up your initial navy, but later you may be able to get back into the water starting with subs under the right conditions.
Good luck and enjoy.