What do you think are the reasons behind Billy's lashing out on Max and his overall negative behaviour towards her? Could we say he was ever honestly protective towards her or just wanted to be the one in control? For example, he decides with whom Max should and shouldn't hang out, but yet insisted she's not his sister and he doesn't really care about her. Before they moved to Indiana, he broke one of her male friends arm. He had a similar approach towards Lucas, but many people say in his case it was racially motivated. What are your thoughts and interpretation on the scene where Billy confronts her for the first time because of him in his car vs Billy threatening her male friends back in California?
Let me set the scene for you.
Billy is 17 years old. He has his own car. He is one year away from freedom. At 18, he can graduate and move out. His dad has had him under his control for years after his mother, who he witnessed his father abusing, left him alone in his "care." What Neil did to her, he does to him.
Then. Neil starts dating a woman who is easily manipulated by bad men - Max's mom. He waits until they're in a committed relationship before he even introduces Billy. Now, Billy has to navigate being the abused child and two more potential victims of Neil - Max and her mother.
But! Neil can't hurt Max without alienating her mother. He doesn't have enough control established over them yet. So, he makes Billy responsible for Max which gives Neil the opportunity to not only punish Billy if he messes up but divide Billy and Max to make it easier to control both of them.
This is how abusers operate. They isolate their victims and make it harder for them to find support and resources. But, Billy and Max and his new wife Susan still have a foothold in California that could jeopardize this. So what does he do? Of course, he plans to uproot them as far away as possible.
Neil picks up the family in Billy's senior year of high school and moves them to Indiana. Billy no longer has friends. He can no longer enjoy his surfing and swimming regularly. He has to drive over 24 hours, likely spending his own money to get to Indiana where all of his things are.
Billy is now much farther away from his mother.
Billy is neither protective of Max, nor does he desire to control her. Neil wants Billy to control her. We see this when Billy defends Max's ability to take care of herself, his warnings about and to Lucas about her relationship with him, and the way Neil literally beats him for not watching her.
Re: The Broken Arm - none of Runaway Max is actually canon to the series, but it may be used to inform their characterization. Billy is like an abused dog, lashing out at everyone - blowing off steam. While his actions are serious, I don't think his motivations are that complex.
Billy is angry. Billy lacks control in his own life. Billy exerts what power and control he does have over others to feel better. This is bad. However, it's also short lived and Billy doesn't make a habit of doing this once the fight at the house is over and they've settled in Hawkins.
The fight at the house is an acute issue, rather than a chronic one. Billy is threatened into bringing Max home. Billy has to forgo his date to find her, which he does at a random house deep in the woods with a bunch of boys. That's weird, and also bad for him who is tasked with keeping Max safe.
I don't discount the racist aspect of Billy targeting Lucas at all, but this is also not without context. We know Neil is a bigot. Not only could Billy internalize this despite having friends who aren't white, but he could view Lucas as another complication in his efforts to keep Neil satisfied.
Billy gets a summer job after high school, presumably saving up money to leave. He teaches swimming lessons to children. He is liked enough by his peers and coworkers. He's still brash, but he's no where near as raw and on edge as before. He really has a chance to make it out of Hawkins.
Until he gets preyed on, assaulted, possessed, and killed. Billy was always playing a losing game. Neil would not let him be free, and when he had the chance at freedom - to live in a context in which he could not only heal but be a better person - it was violently ripped from him.
It is very easy to understand why he can't stand Max when we understand what Billy has gone through and none of it is because Max did anything wrong. She didn't. She was an obstacle for him, though, in his efforts to survive and Neil intentionally made it impossible for them to truly bond.