I have an 11 year old son, and I have never not reminded him to shower and brush teeth (hair is not an issue). He protests, but it's not negotiable.
So that doesn't really answer your question, as I've just continued to make sure he does it. I don't know that he would do it if I didn't insist. I have an 8 year old. Since I make it part of our daily routine (and I think it's perfectly age appropriate for me to be on top of the 8 year old), I haven't really tested the theory of whether or not he (the 11 year old) would brush teeth or bath on his own.
I can tell you that he has always put on clean underwear after a shower or bath, so that's not something I've ever dealt with. You mentioned concerns that your daughter wears the same underwear and socks for multiple days. Do your think she's putting on the old after taking a shower? Or do you think she's simply not taking a shower?
I'm thinking there are a couple of ways you could handle this. You could simply begin making it a part of your nightly routine - insisting and following up on showering and brushing teeth. That's putting it on you and not her, so it might not teacher her to take responsibility. Then again, if it's just what you do everyday, she probably will take that with her as she grows.
You could, instead, let her know what the consequences will be if she doesn't do these things on her own. Let her know that this is her responsibility, that you will be checking (without reminding) and that there will be consequences (no tv, phone, tablet, whatever) if she doesn't.
I guess I'm just running with the theory that if it's part of our daily routine, even if my husband or I are reminding the boys, that they will just always see this as what they do every night before they go to bed (and will continue to do so even after they move out).