Regarding possible hirsutism (the new hair), abnormal cortisol levels can create this state.
Corticosteroid medications mimic a natural hormone called cortisol, and cortisol is produced in the adrenal glands which also produce sex hormones (as do ovaries and testes). When one level is altered sometimes it alters the production in another layer of the adrenal gland.
You may want to consider adrenal function testing, or just talk with an endocrinologist about the excema and various infections and the meds that are all being prescribed, because the endocrine system governs the immune system. We need healthy levels of hormones in order to have normal immune function.
Small physiologic dosages of corticosteroids can actually improve excema and immune function long term in individuals who have a deficiency of cortisol, but when dosages are excessive it can deregulate immune function, just as much as too little cortisol can.
People (and animals) who do not produce adeqate cortisol, or actually produce defective cortisol, that the body is unable to utilize, are more susceptable to infections, and they tend to respond favorably to small doses of corticosteroids, namely cortisone and hydrocortisone (those most like the natural hormone cortisol). It is fulfilling a deficiency of an essential hormone. However, a stigma surrounds corticosteriod pharmacauticals, and rightfully so. They are often misused. Most doctors never do any baseline testing to see what the body is already producing, before adding more to it in the way of administering pharmaceuticals, be it injection, cream, pill, etc. Therefore, doses are usually too high and when there is excessive cortisol, it also deregulates immune function, just as a deficiency does. Unfortunately most doctors see corticosteroids as a quick fix without ever seeking the root of the problem.
There was an endocrinologist, award winning, published in New England Journal of Medicine, and he was a research fellow at Harvard. His name is Dr. William Mck Jefferies. I read his book, Safe Uses of Cortisol. He studied cortisone for decades. Basically the pharmaceuticals that we have now are very 'suped' up versions of cortisone and very unlike the natural hormone. Where as Cortisone and Hydrocortisone are more so like Cortisol. However, even excessive dosages of these can cause side effects. I would reccomend you google Dr. McK Jefferies work, there is a lot available online.
Another doctor, Dr. Alfred Plechner, has done a lot of research on immune function in relation to endocrine function. He discovered in animals basically the same thing that Dr. Jefferies did, small 'physiologic' dosages of cortisone, improve long term immune function in individuals who lack sufficient cortisol production OR whose bodies produce 'defective' cortisol hormone. So by replacing what the body does not produce, similar to how people replace thyroid hormone in those with hypothyroidism or insulin in diabetics, it fulfills a deficiency so that the body functions more normally. The key though is to only replace what is needed, so physiologic not pharmacologic dosages. Pharmacologics dosages are excessive, and they are the most common dosages prescribed today. By the way, Dr. Alred Plechner is a veterinarian, whose work is recognized by the medical community. He even speaks at medical conferences, and has his research at the Broda O. Barnes Medical Research Foundation. Dr. William McK Jefferies work is also there.