This usage corresponds to the one described in section C.c. of the TLF’s entry for Qualité. Roughly here the qualité would be the ground that allowed your advisor to supervise your PhD. But it could be a title, a status, a position… it is not obvious to tell what exactly qualité is referring to.
In this case, the denomination of his/her position is most probably expected. In universities, usual options are Professeur (also used for any foreign teaching position) or Directeur de recherches. But it could also be Maître de conférences or Chargé de recherche, etc., in special cases.
Yes and either one of the two answers below is expected:
- Doctor, HDR
In the latter, HDR means that he/she can supervise researches.
There is no relation with any civil title. It is more in the meaning of
 HDR is the highest University french diploma. Usually, people obtain it after their Ph.D. People having an HDR can be
Directeur de thèse. They can also apply on
Professor position, provided that they get qualified first.