Best thing to do with any Arena questions is head over to Arena FAQ, or RPG Ranked, or just general Arena Chat.
However.
1. Usually you're fighting to decide that, it is essentially a competition, which rewards originality and thinking on your feet, as well as detailed and realistic attacks. Usually I've won fights by beating my opponent into submission, but in some tournaments the winner has been decided by a judge.
2. Depends, the whole point of the Arena HQ is establishing a multi-verse where rivalries can form, but many other people fight without a greater story connection. Personally I like to establish rivalries and interconnected stories with other fighters.
3. That depends on how you want to fight, there is an established system in place where you essentially fight in 'turns' which involve an attack, a defence, movement, etc. When someone launches an attack at you, there are three initial choices based off logic. One, if your character can't avoid an attack because of the situation you take a 'hit' where you describe what your opponents attempted attack does to you. Two, you avoid the attack, which is usually the safest option but not always possible. Three, you defend yourself, raise a shield, parry, whatever you want. After this you have the chance to either counter attack, where you launch your own attack, or allow proceedings to reset. Ultimately it all depends on the flow of combat.
If you're still interested you can start learning some of the more complicated stuff like preps and how magic works into this system, but like I said I'd recommend checking out that FAQ or Arena Chat, there are plenty of people there who can answer any questions you have. Also, check out some fights and you'll see how they go down.
Hope you give it a try, see you around.
