No racism would also impact things like cultural beauty standards. The prevalence of hair straightening is due to straight European hair being set as the beauty standard. It means in certain areas, like business, hair that isn't straight is expected to be straightened or it's not deemed professional. In your class-based world, this'd be irrelevant as long as the hair was well-maintained.
This is true and an important point, but I also want to point out the flip side: there is a tendency to assume that if a non-Western culture happens to have standards of beauty or use cosmetics in a way that remotely resemble Caucasian features, then clearly that means they want to be Caucasian, even if those practices predate European contact and have nothing whatsoever to do with Europeans.
No, if a culture happens to use white make-up for some purpose, that does not mean they want to be white.