No. The Doctor becomes Engaged when they enter Base contact with an enemy model, regardless if conscious or otherwise, and Doctor is not an allowed skill to declare while Engaged. http://infinitythewiki.com/en/Engaged Edit: look at that, there was a page 2 :p Ah well, now you got it answered twice.
Asked the question about healing the Designated Target (for your enemy, but still your own HVT) during the Interplanetario stream. There has been said that you can't heal it. But not so sure about it.