Dye for Diapers
I would not recommend the Dylon Dyes. For sure go with the Dharma Trading Company Dyes. They are beautiful and they bond chemically to the diapers so that the dye doesn't come off on your baby. It is permanent and will not fade! If you have any questions about Dhama Dyes, let me know. I got the dye specifically to dye my diapers.
I use these dyes often for my toymaking work.
The regular procion dyes from Dharma are really safe; I wouldn't seek natural dyes unless you simply like the idea of them or their soft look for your own pleasure. The fixatives you'd use with natural dyes are as far as I know no less hazardous and can sometimes be more hazardous than the pigments in procion. They are less reliable and less strong in color, though soft colors could be pretty on dipes. The only concern with Procion dyes is that they are a dust and could irritate if you breathe it, it's not a toxic dust though.
The only problem with tie-dying refolds is that they are so thick it really limits what you can do in terms of folding and tying them to make patterns. That chunkiness makes it hard. I'd probably go with solid colors just because of that.