Hi! For kids, I like using medium tubs for storage. Smaller tubs for larger parts, and shoebox tubs for tiny parts and mini figs.
To help with the floor clutter, I converted an old wooden train table (check fb marketplace) to build. Instead of the storage bins, I was able to use two shoebox tubs and four of the next size up so stuff can be put away and covered. Then I found a medium tub that fit under the table for a big parts bucket.
This gives a place for builds to be put when play stops without worrying about them getting busted by being put in a large tub
I only ever saved boxes for sets that were well over $100. Everything else was recycled. Instructions are stored away in a filing cabinet drawer.
Edit: I just cross-posted something that was submitted to my old sub on reddit on storing instructions.