senperfect

^ In addition, in all such matters, involve the instructor of record. The professor in charge of the course should know of any issue serious enough to warrant removing a student from class (name calling would be included for me, argumentative student in discussions would not, but that is a judgment call). If it were me, in cases that don't involve safety I would do my best to ignore the bad behavior and steer the class back in a productive direction; engaging a student like that could be dangerous in that you as instructor often can't actually do much in the moment and you don't want to give the student too much power (e.g. refusing to leave and causing a scene, taking time away from class). It's a judgment call. In many cases, the right course of action would be to turn matters over to the professor and not deal with it yourself. The power differential between prof and student and the experience will usually mean it'll be easier for the prof to deal with it than for the TA. 

I think a lot of the specifics will depend on your school's code of conduct and other policies. Although I have never had to do it myself, in general, I think a TA should be able to dismiss a student from their class if they are calling the TA a profanity (especially if there is unlawful harassment along with it) or if they are disrupting the learning environment of other students. Also, as an a TA, I think you should have the right to not deal with abusive students in your work environment (whether you are protected by labor laws will depend on whether you are treated as an employee, though). Finally, if you are responsible for students in a lab setting, I think violation of lab safety (the loud music example) is reasonable grounds to remove a student from a lab.
For the serious infractions like this, I would personally lean towards asking the student to leave and deal with whether or not I was allowed to do that / should have done that later. I would generally prefer to take immediate action that eliminates the safety risk (if one exists) and remove disruptions to other students' learning. Then, as necessary, I would go to the appropriate offices on campus after the fact to figure out what to do next, if anything needs to be done, so that the incident doesn't happen again. But the protocol for each school will vary with each school's policies! You generally learn this during orientation or some kind of TA training.
For the example of the student derailing the class discussion, or other smaller disruptions, I don't think dismissing the student is the right response. Usually, during TA training, you are taught some strategies to keep the class on task and redirecting the discussion to the relevant topic. I find that usually, a reminder that X is not the topic for discussion today and then disengaging from the conversation works well for stopping discussion on an irrelevant topic. There is a pitfall to this approach though, because depending on the course material, while the instructor might expect students to approach the topic in an unbiased manner, the truth is that we are all humans and it's not easy (or even better, necessarily) to approach everything from a disinterested view. I think it's important to create a classroom climate that encourages learning and growth. So, if a contentious topic like climate change comes up during a class and if it's somewhat relevant to the topic (even if a little tangential) then maybe a good thing to do would be to schedule a discussion on the topic for the class in the future. I'd explain that I have already prepared material for today's class but we will try to work in the climate change discussion later in the term. And then follow through on it if you can find a way to make it work with the course learning goals (usually possible and probably better to engage with an example/topic that your students care about instead of an arbitrary one that you picked). If you expect the discussion will be heated, be sure to set "ground rules" for the discussion at the start. 

Are the students afraid you can't teach, or are you afraid? I feel like it's a very strange way to phrase the question, and it's left me unsure as to what you're really asking.