Do birds worry about their hair? Well… They dont have hair, so probably not.

But do birds worry about their plumage?
Yeah, probably they do.
Because there are many examples of birds using their plumage to attract mates, best known is probably the peacock.

In short, my thinking is that you need sexual procreation and a certain minimum intelligence to care for their outward appearance (e.g. I don’t think ants care for outward looks). So probably all mammals and a lot of birds definitely do care “about their hair”. But insects and fish do not.