Hello,
Welcome to the game!
It was made to get the player familiar with the different type of units in the game without having him feeling overwhelmed :)
The carrier in this scenario has absolutely no purpose, since you have airfields nearby. But if you start playing bigger scenarios, or scenarios without any airfields nearby, you will be happy to have a carrier with you to refuel, repair, and rearm your carrier planes.
Aircraft carriers provide the same services as regular airfields, except they only work with compatible planes (carrier compatible planes). Look for planes having an anchor icon. These are carrier compatible.
Yes, factories and refineries both produce income and will boost your economy.