First of all, cortisone injections wouldn't thin your skin, certainly not permenently. Second, the mites of canine mange don't adapt to humans well, so they cause just a minor and self-limited rash. I guess it's possible that the Elimite cleaned up the remaining mites, but it isn't absolutely certain that it was mites.
I'm also not sure that the veins yuou can see through your skin are clearly abnormal. Is it possible that you are just noticing them more now? (I ask because seeing veins clearly isn't steroid side-effect I'm familar with.)
In any case, whatever may be steroid-related will go away with time, so you don't need to worry about permanent damage.
Take care.
Dr. Rockoff
Dr. R