Hi! First, some people stamp without using an ink pad, but I find that the coloring ends up much more evenly distributed on the stamp if you use a pad (as opposed to painting or sponging the coloring on the stamp). And, yes, I like to dedicate one color to each pad. I suppose you could wash out the pads between color uses, but the pads would take a while to dry, and so I simply close my pads and store them in baggies so nothing gets into the food coloring.
As for "ink", just to be clear: I only use edible food coloring as ink on cookies, and any brand and many types of food coloring (gel, paste, liqua-gel, airbrush) will work. However, I vastly prefer liqua-gel or gel coloring to airbrush coloring. It is thicker and so is less likely to leave you with a smudged imprint, though it may take a bit longer to dry. I normally use my Chefmaster liqua-gel coloring that I use to tint my royal icing.