The first rule of HR is that you must apply the rules equally to everyone. The second rule of HR is that you need a written handbook of employee rules (assuming you are a larger company) -- plus the addition of having a clear social media policy in the modern era. The third rule of HR is that everyone can define themselves as a federally protected minority in some manner when a case gets to court -- everyone is a minority either based on gender, race, religion, or some disability (including imagined disabilities). From a legal perspective if you don't apply the same policies equally to all of their employees then you are going to lose in court. If Lucasfilm fired Gina Carano stating it was due to her social media posts and took no action against a male co-star with even worse and more numerous social media posts --- then Lucasfilm better be ready with a payout. However this being Hollywood I expect that Lucasfilm has already provided Gina Carano with a large cash settlement already (or shortly) with a signed agreement in place for her not to sue them.
Political affiliation isn't a protected class. Your "sexism" hypothetical doesn't pass the sniff test. Good luck proving a male cast member making her verbatim statements wouldn't get shitcanned.
True in every state under Federal law when it comes to the court system. Plus enforcement by the federal EEOC which fines employers for discriminatory practices. This includes the at-will state where I live , North Carolina. Every tech/pharma company in RTP in NC is very careful to be sure to apply the same policies equally to all of their employees -- this comes to social media policies as well. All it takes is an employee to point to an example of where they were disciplined/terminated for an offense and a fellow employee was not disciplined/terminated for a similar offense to be on the wrong side of the EEOC and a court case.
Is @gwb-trading now for socialism? Is he against free-market businesses hiring and firing whomever they want, as long as they don't violate Title VII? Is @gwb-trading a hypocrite? Why won't @gwb-trading hire that bitch and make his own movie? Isn't that what capitalism is all about? But no! He wants socialism policies to be put in place to protect his favorite actors from the right of their employers to fire/hire at will. I bet he wouldn't want such restrictions against himself in relation to his own employees.