Sorry it has taken me so long to reply. Our Internet was down for several days and for some reason my Curezone mail has been going into spam....
Iodine will stain the skin for a while. It goes away pretty quickly, though. (i.e. if you get surgery and they use Betadine to disinfect, it usually doesn't last for more than a day or so!) I use the Lugol's solution for supplementation, but I would not use it on broken skin unless I had added it to an oil, for instance. I do keep a bottle of coconut oil, into which I have added some iodine, in the bathroom for face cleaning -- maybe because it's in oil, maybe because I only use a couple of drops per tablespoon, but at any rate, that has not stained my skin -- it just wipes off.
I would definitely check the Iodine concentration in oil on a non-facial body part before using it on my face, though. You could end up looking like an Ooompah-Loompah!
For supplementing, I will sometimes "paint" my daughters' feet or legs with a couple of drops of Lugol's. The skin will absorb it, although some will oxidize into the air, and some will just rub off. If you look on the Iodine forum, you will find that many people prefer to paint Iodine on rather than ingest it.
Hope this is helpful....and that you all are well!