If I'm playing as Popo (ice climbers), any hits on Nana will not affect my percentage. Does that mean that Nana has her own percentage score? If she does, is there a way to tell what it is?
Yes, Nana has her own percentage score. As it goes up, she'll become easier and easier for your opponents to kill, just like your own.
No, you can't tell what it is. Just pay close attention. If you both get hit by the same attack, judge the difference in distance between how far you went and how far she went and you can estimate based off that.