I would make a loose paste of baking soda and water for both applications. Rub with a coarse, but soft cloth like a washcloth. Do small bits at a time and work in a circular motion, moving down the polish until it comes off.
You can also try using a fine grade sandpaper, the kind that works on removing the mud from drywall when prepping for paint.
If that doesn't work, try non-acetone nail polish. Hopefully, the polish isn't made with gel, as are many today. I wouldn't put that stuff on my nails if you gave it to me!
As far as the carpet goes, I would do what I've done in the past when I've spilled hot candle wax: pour some boiling water on the stain, then take a razor blade and scoop the melted substance up. Follow with dipping a cloth in a baking soda paste or diluted white vinegar. One thing you need to watch about using baking soda is it's a natural whitener, so if your carpet is dark, you might want to go with diluted white vinegar.
If and when you do paint, apply a paint made for bathrooms and kitchens that can easily be wiped clean and resists stains.
Also, keep your nail polish out of reach of your children. Keep your bedroom door shut and let them know that's YOUR private space!