Hmmm lets see here. I don’t think I’ve ever seen that before on a cactus. Tbh I think the main cactus could be rotten. Something strange is definitely going on. Take it out and remove all the hard soil. Then you will be able to see what’s going on. From here it looks like it’s dying. That white stuff could be mold.

The little babies could be removed, placed on a fast draining mix, and some will form roots. Remove the healthiest and biggest. The mix should be 60% rinsed strained perlite or pumice and the rest cheap cactus mix. While you’re waiting for roots no water, keep away from harsh sun. When roots form you can begin to water and slowly acclimate to stronger sun.

Good for your daughter picking a spikey cactus and not the “pretty” succulents. I’m a cactus girl too;). The one she has is an Echinobivia Rainbow Burst. They’re common so you guys should be able to find a new one if this one doesn’t work out. The flowers can vary from each Rainbow Burst.

The Rainbow Burst is the guy with the orange flowers.

Goodluck. Let your daughter know that it’s not her fault. You guys have only had it for a month so the downward spiral started way before you bought it;). Here’s some info;) on that genus of cactus.