First, when I bath I put a towel in the sink so they don't slip and it slowly lets water build up so their feet and bottoms soak. Luke warm water and a sprayer and go really slow working from the rear up. Mine don't panic. First bath they are a little tense but then they start to realize the water feels good. Baby shampoo is the best too and I tip the head beach and spray so water doesn't run in the nostrils.
As for the dye, if you search eBay, they sell dye for dogs and it's non toxic. You bath them and then put the dye in while they are wet. Let it set 15 to 20 minutes. While they sit there with the dye on them, I wrap em in a wet warm towel and spray it here and there with warm water so they dont get a chill. When it's done, rinse with warm water til water runs clear, towel them off and blow dry. Mine LOVE the warm air from the blow dryer. They start yawning and lay down and fall asleep. It's really cute. Some people say dying is cruel but mine love it and they get all clean and depooped. It's only cruel if you hurt them or they just truely hate the bath and panic every time. I have yet to have one that is terrified but mine get handled and picked up a lot.