is it okay to dye things in a pot that will later be used to cook things to eat from?

Its probably not a good idea in general.

It depends on what kind of dyes you are using: Food color dyes such as are used for Easter eggs are usually not toxic but some clothes dyes might be. At the very least, wash the pot well with bleach. But I wouldn't do it, I would use the kitchen floor bucket instead.

I'd say no, unless the dye is made from pure beetroot. Commercial dyes are made out of a variety of chemicals. Get a pot from a second hand store and use it for sole purpose only.