You might want to consider getting a karafun subscription and run it on a device you already own i.e. tablet, phone, Smart TV, laptop.

https://www.karafun.com/apps/

Karafun has a parental control option which will remove the explicit lyric songs.

https://www.karafun.com/blog/165-parental-control-now-on-karafun.html

Karafun does have playlists for Nursery rhymes and Disney songs

https://www.karafun.com/karaoke/playlist/disney-55/

https://www.karafun.com/karaoke/playlist/nursery-rhymes-15/

For microphones you might want to consider something simple like this

https://smile.amazon.com/BONAOK-Bluetooth-Microphone-Christmas-Smartphone/dp/B075CKXXKH

Short of some subscription service you will have issues trying to meet your requirements.

You can buy a cheap karaoke machine off of Amazon but you'll still have the issue of getting songs since most machines don't come with any.

This machine DOES include some songs but I have no personal experience with it or how good the song quality is.

https://smile.amazon.com/Karaoke-USA-GF842-Machine-Bluetooth/dp/B01FW4BW2I

You will still need to buy songs. Amazon sells karaoke discs that should play on this device

https://www.amazon.com/Childrens-Favorite-Songs-Disney-Karaoke/dp/B005CA4E4Q

You can download packs of kids songs from places like here that should play on certain karaoke machines or other devices that support the karaoke format.

https://karaoke.stingray.com/store/search/packs?searchText=kids