You probably know it's not so much about the stain. Or even your parenting skills. No matter what the circumstances, they've been through a huge change and that's always hard to adjust to.
You might try this for info about the coffee stain:
HowStuffWorks "How to Remove Coffee Stains"
Why not involve your sister in the decision? You could explain to her that you know it's her favorite and you really want to fix it right. If she's included in researching what to do and carrying it out, she can't exactly blame you. Plus at that age it wouldn't hurt her to learn to be responsible for her own laundry.
Makeup on all the bath towels? Why is she using bath towels to wipe her face? You could try those disposable make up remover wipes and tell her a) it will save the towels and b) it will be better for her youthful skin.
For sweat stains:
HowStuffWorks "How to Remove Sweat Stains: Tips and Guidelines"
Come to think of it, I went through a period where I was very angry at my wonderful parents when I learned that they didn't have all the answers. My mom's solution was "let's sit down and figure this out together." It worked pretty well, and that was even before the internet.