"Squeeze the sponge out" would mean you squeeze the sponge until there is no water left. It's a completed action. "Wipe" is one method of drying. Hair-dryers are another. You could wipe using something wet, or using something dry. "Dry the baby's skin" implies that you use a towel. Hang the towel "up" means "up off the floor". Usually, things we hang up are eye-level or higher, but if you're confused about a knee-high drying rack, then just use "hang the towel on the drying rack". We'd still want to know where the towel goes.

Thank you very much!!!

you can say "hang the towel up to dry", it doesn't matter what level you are hanging the towel. "Hang up" is the act of hanging the towel, wherever it is. Similar English words are: "close up". The shop will close up at 2pm. This means the shop will close at 2pm. "type up". Please type up this letter. This means please type the letter. "run up". Can you run up to the shop for me? This means go to the shop for me. "clean up". I am going to clean up the room. This means I am going to clean the room. "wipe up". She will wipe up the mess. This means she will wipe away the mess with a cloth. If you are hanging the towel outside in the sun, you could say "hang the towel out to dry". Wipe / Dry - you can wipe with a cloth, but not necessarily to dry something. If you wipe some spilled food, you might use a wet cloth or a dry cloth. But if you dry something, the purpose is to make it dry. So you might wipe the baby's skin with skin cream, but you would dry the baby with a dry towel. I hope this isn't too confusing!! :-)

Can I say "hang the towel UP to dry" when I want my daughter to hang it from a drying rack that is at the level of my knees and at the level of her chest? Or it's better to not use the word "up" in this case?

Hang...up. Because the towel is ``up``, above the floor. BUT the sentence still makes sense if one says ``hang the towel to dry``