Those are very good questions and I'll only be able to offer an opinion as the research is not there to support anything more. The term "autoimmune" is an interesting one from a Chinese Medicine perspective (and from a "westerners" perspective as a Chinese Medicine practitioner). The general idea in the west is if we are sick build ourselves up. This is the flawed logic behind the very profitable supplements market and a seemingly cultural logical flaw in many western Chinese medicine practitioners. What I mean by this is there is too strong of a tendency from our collective unconscious and cultural logic to tonify, tonify, tonify.
To complicate matters the term "autoimmune" simply means an out of control immune system doing things it shouldn't. The result of many autoimmune conditions can be weakness, etc. (among, of course, far worse things). As a practitioner you (and I mean a general you - not "you" specifically) need to decide which side of that equation you are going to be on. Do you want to (a) regulate the immune activity or (b) deal with the repercussions of that activity. Generally from a Chinese Medicine perspective you want (a) - the root.
Personally I would step back and think the case through more (perhaps you have, but your question is missing quite a bit of information I would need to offer more information). First, what condition does she have specifically? And what is her overall Chinese Medicine diagnosis - does she have signs of deficiency? Autoimmunity can (not always) be related to yin deficiency which leads to a hyperactivity in body functions from chronic hyperstimulation. If you look at the major autoimmune conditions from a Chinese perspective yin deficiency stands out very often. Do you moxa someone who is yin deficient? perhaps, but only with a very clear purpose (other than simply "building" immunity).
Another avenue of thought on this issue is whether "tonifying" immunity from a Chinese perspective can make autoimmune conditions worse by adding more fuel to the fire. The clinical studies haven't been done to support this idea one way or the other. From a Chinese Medicine perspective, then, a clear diagnosis is important and from that you can deduce whether or not moxa would be appropriate.
To specifically answer your questions, I have included them below:
What do you think? Have I harmed my patient by doing moxa? I did it one time only on her down the UB mai. I will not do it any more, but am searching for specific details on:</li>
No - one treatment of moxa is not going to help or harm anyone in any significant way.
1. how many days is moxa done to boost the WBC count?</li>
From clinical studies it happens almost immediately but is relatively short lived - Similar to the effects of practicing Tai Chi or Qi Gong. In other words, it would have to be done consistently to make systemic changes in the body. Moxa except for healing trauma and other local uses is generally fairly slow - particularly when compared to internal herbal medicines. This often makes it a safer way to tonify particular conditions.
2. how many cones are burned at each treatment?</li>
There are a hosts of theories on this and it would depend greatly on the patients case - what points you are using - and the season. If you have someone who is significantly deficient you could easily do 6-9 cones at particular points daily. For general vitality many practitioners (and patients) do daily or perhaps 2-3 days/week of moxibustion at ST 36 - perhaps just 3 cones.
You may wan to read the relevant sections within our Japanese theory section:
Chinetsukyu - Cone Moxa Theory
Okyu - Thread Moxa Theory
3. can moxa encourage the body to be sick rather than strengthening it?</li>
See the above discussion on autoimmunity. My general sense is no, but it can add too much heat into the body in people with hot/excess/empty heat patterns and this can be damaging. Again, you are generally using moxa to warm and tonify - so in anyone who has empty heat from yin deficiency, excess heat, it's the middle of summer, they are agitated, etc. you need to use it sparingly if at all.
Now this doesn't quite include Japanese theories of okyu (thread moxa) which are more complicated as far as their usage in a wider variety of cases.
Hope this helps! The brief sentence answer to your question is to use moxa on your patient if it is clinically appropriate for her underlying tcm diagnosis and for no other reason.