You don’t lose your right to free speech on the job. It is Obama types that want free speech zones and restricted speech everywhere else. I don’t have to agree with someone to defend their right to free speech.
Yes, you do. When you are representing your employer, you do what they pay you to do, not anything else.
I expect the waitress at Chili’s to serve me cheese fries, not lectures on politics.
I expect NFL players to play football.