Without much context on the lab manager's behavior it's hard to know exactly what's going on here, so I'll start by saying it is obviously never OK to create a hostile work environment, and if you feel you are being harassed or the lab manager has created a hostile environment based on gender, you should know your Title IX rights and how to report the problem. Regardless, you should absolutely have control over what you are called-- have you firmly but politely asked the lab manager to not call you "babe"? I also agree some of the words you report him/her using, such as retarded, are not the most sensitive terms and should be avoided.
However, it sounds possible here is that you are mostly just experiencing a difference in lab cultures, with more joking around and informality than you are used to. Do others in lab meeting join in in joking around and using informal language? In my postdoc lab we were very informal together, and you would have definitely heard the occasional swear word in meeting. I don't think that made us unprofessional, context is important-- we would of course not have done that in front of participants or outside colleagues. Some lab cultures are much more formal and perhaps you are coming from one of those and just need to adapt to a less formal dynamic. But again, without much context, it is impossible to know if there is instead a bigger problem here.
Finally, I agree with greyscale that you need to be careful of seeming like tasks are beneath you, even as you work to protect your time and clarify your role in the lab. That's what I meant about being a team playing when you can. I think a good rule of thumb is if it would take you longer to explain why you shouldn't have to do something than to actually do it you should just do it (this would apply to greyscale's taking out the bio trash example). There are also certain tasks that everyone needs to pitch in on. For example, in my postdoc lab we all took a certain number of evening and weekend participant slots, regardless of "rank", and coincident with my leaving for my new position, the rest the lab was moving across the country, and we all spend the last few weeks of my time there sorting, packing and recycling. Did I need a PhD to do that work? No, but it needed to be done and we all needed to help make that happen. However, other types of tasks that were ongoing time-consuming responsibilities, like scheduling participants and data entry, were clearly not good uses of my time and could be done by others, so I didn't do those things and spent time writing etc. instead. Well-run labs tend to explicitly talk through how to divide up tasks so as to make the best use of everyone's time. It sounds like your lab is well overdue for such a discussion.
Good luck and keep us posted on how the discussion with your PI goes.