Testing homeopathy and single blind tests

Is it possible to test homeopathy? Adherents often say "no," because it's impossible to perform a double blind test.

Is it possible to test the influence of a single blind test?

Regarding homeopathy: Perhaps one cannot prove that it does work, but I think one could certainly prove it doesn't work. The standard claim is that it cannot be tested at all, because one cannot perform a double blind test (i.e., a test where the practitioner doesn't know which treatment he/she is administering).

The reason I claim that one could show that it does not work (depending on experimental results) is that I presume that (1) if the practitioner knows he is administering real drug, he will appear more confident, etc., and that (2) this will tend to improve patient outcomes. Inversely, I presume that (3) if the practitioner knows he is administering placebo, he will appear less confident, and that (4) this will tend to hurt patient outcomes. As a result, if homeopathy outperforms placebo, this might just be because the doctors were more confident. But if homeopathy fails to outperform placebo, despite having the advantage of being administered by doctors who were particularly confident and who believed in the treatment, this would be compelling evidence against homeopathy's efficacy.

But can we test these presumptions (#2 and #4)? Although this might be impossible in the case of homeopathy, I think that with ordinary drugs the influence of a single blind test could be tested as follows:
For (2): provide some doctors real drug and tell them it's real drug; and provide other doctors real drug and tell them it's placebo.
For (4): provide some doctors placebo and tell them it's real drug; and provide other doctors placebo and tell them it's placebo.

It would be interesting to see how the doctor's state of mind while administering treatment influences patient outcomes. Have tests like these already been performed? If so, what has been the result?