In my opinion, Batman would probably win. Batman has a vast array of high - tech gadgets and is a master of many forms of combat. He's also a brilliant strategist. Green Arrow, while an excellent archer and fighter, doesn't have the same level of gadgetry and has been shown to be a bit more impulsive at times. But it really depends on the circumstances of the fight in the fanfiction.
A really cool storyline might be a situation where they are both framed for a crime they didn't commit. They have to work together to clear their names while also dealing with their own personal issues. Batman might be suspicious of Green Arrow at first, and vice versa. This would create a lot of tension and drama. Also, a fanfiction where they are transported to an alternate universe and have to find their way back could be exciting. They would face new challenges and enemies that test their abilities in different ways.
Green Arrow isn't typically portrayed as an asshole. He's often a hero with a strong sense of justice, but like any character, he has his flaws and moments of difficult choices.
Yes, it can be. It often gives enough background within the story so that new readers can follow along. They'll quickly get a sense of who Green Arrow is and what he stands for.