I've been trying to remove all the "I am currently reading", "favorite books" and "favorite authors" from my profile and it won't update. I remove them and then click on 'edit'. I don't see anywhere to click on 'save'. Am I missing something here? 

I don't think it will keep it empty.  If you don't choose what to put in there yourself then it picks stuff from your transactions/bookshelf/wishlit.

Mary is correct.  Unless you specifically put something there, the system populates those categories based on your account.

You can go to settings and choose privacy controls and then change the view for those items to "only me."  That's about all you can do.  At least other people won't see them when looking at your profile.

Mary is correct.  Unless you put books there yourself, they will autofill with books in your other lists (BIR, TBR, etc).

Something else to keep in mind: the browser is sometimes the problem when trying to update, not the system.  Internet Explorer in particular, I have found, has a hard time with the code on this site sometimes.