Question 1: Do not quote me on this but I was looking into rust removers earlier in the year and from the very basic research and reading I have done, the rust remover leaves a layer of iron phosphate particles on the surface to reduce the chance of rust from re-forming again. Some removers add a small amount of iron phosphate to reduce loss if surface iron and increase rust resistence.
I do not know if the iron phosphate will come off on the cloth, though I suspect it would. The light yellow colour might be some kind of iron oxide being removed.
Question 2: Water wash then dry it very well, then use isopropyl alcohol (rubbing alcohol) might help though. Some rust removers use isopropyl alcohol and 2-butoxyethanol though I do not know why the 2-butoxyethanol is used, though I think it is for dirt removal so all this should just wash/wipe away/evaporate away.
Question 3: No idea really. All rust removers I have come across contain phosphoric acid and surfactants.