I wouldn't be concerned with ketosis being interfered with over a bit of iodine, but rather autolysis and the body's swiftest and most perfect transition towards balance, homeostasis and optimal functioning. My recommendation here would be to brush your teeth only with water using any brush you'd like. It works wonders for oral hygiene including breath and you can have this quickly verified through a trusted loved one. All of your symptoms seem quite normal in a fast and all i know to do for the hot/cold issue is to continually make the necessary adjustments to keep yourself comfortable.