Yes, I did stop Iodine for several months but I found out the hard way that once you trigger this it makes no difference. Let me try and explain it a bit more clearly for you. There are 3 things you need to trigger autoimmunity or trigger an autoimmune attack if it is sub clinical.
1 A genetic disposition, so any auto immune problems in close relatives would point to this and you have alopecia anyway.
2 A leaky gut which you already have with rosacea.
3 An environmental trigger.
This is where the Iodine comes in. It is not the direct environmental trigger but it forces a detox which displaces hallides, latent viruses and heavy metals which are the environmental triggers which is what you are seeing. Alopecia is an autoimmune disease as well, and you obviously have that so the chances are you will have hashi's if not now probably in the future because it is the most common auto immune condition by a long chalk. You need to deal with it, as having one puts you at risk of developing more. Hashimoto's is not a thyroid disease, it is an auto immune condition and the most important thing to do first is to heal and seal your leaky gut to bring down the inflammation - you will get nowhere without addressing this first. There is masses of information on how to do this on the net so I won't go into detail. I hope this makes sense?