I believe Auron's look is based around the premise of something called a Ronin, a certain kind of Japanese warrior. Someone else would know more about that.
I don't know much about the term other than it was used in an old X-Files episode to describe "a samurai without a master." In other words, a mercenary for hire. I'm not sure how that fits in with Auron's look though

As for the arm, although it looks like its in a sling, he does use it now and then, and it seems to work fine. I don't know if this means that its injured but he can use it if necessary. Either way, I would guess that any injury he has, including his arm, is a result of his confrontation with Yunalesca.