no, the different slips should not bleed into one another unless you layer them when they are sloppy wet - and in that case it should be pretty obvious when you're witnessing the previous color bleeding into the 2nd bucket while you dip. simply dip the object, let it dry to around leather-hard or slightly wetter (at least enough to lose its glossy sheen), then dip your next layer(s) and repeat. one layer should net you a thickness roughly 1/16" thick is my guess (on a non-porous object), but truly depends on viscosity of the slip. you can also do multiple layers of each color so they're thicker if you need it.
the main reasons why i suggested food coloring is because the stains or oxides are pricey, but also because it sounds like you're going to be wasting the materials by hosing them off...and then you have the environmental factor depending on where your materials end up once hosed off. clay materials should be fine to wash off into environment, but once you start adding in metallic oxides in concentration we're talking about something different.