What drives all the squabbles about free will and determinism? Is it anything more than a desire to reward and to punish, especially to punish?

What you're asking is really an empirical, psychological question -- What motivates the various sides in a particular controversy? -- rather than a question that philosophers, as such, are well-equipped to answer. But I'll hazard an answer anyway. Take some carefully, even painstakingly, considered decision, such as U.S. president Obama's decision to order the May 2011 hit on Osama bin Laden. If that decision wasn't one for which the agent is morally responsible -- i.e., morally liable to praise or blame -- then I don't know what could be. But according to the incompatibilist side of the debate, if determinism is true then Obama bears no more responsibility for his decision than someone high on PCP bears for his/her decision to try to fly from the roof of an apartment building. According to incompatibilism, if determinism is true then all decisions are equally unfree, equally lacking in responsibility, regardless of how sober, well-informed, and deliberate the decision-maker is. The philosophical...

Is one ever morally justified to beat someone up for making sexist and/or misogynist comments (this is a serious question)?

I can't see how it could be. Beating up someone for making sexist or misogynist comments is using physical violence to punish the commenter. That seems like literally the wrong type of reaction to merely verbal misconduct. (Notice that we don't punish slander or libel that way.) It's something like a category mistake , in addition to being a moral mistake.