Published and unpublished dissertation; DOI assignment
You should avoid citing it twice, because it's the same work (think of it as a book published by two different publishers - you wouldn't cite both versions). Whether you prefer to cite it with a ProQuest DOI or with a link to the university repository is up to you, though possibly a given journal may have an opinion on which they prefer. Depending on how formal the citation style is, you could do something like:


Bleckh, A (2015). ProQuest citation; DOI. Copy available from [repository]


which would let you use both access methods.

For DOIs, it's unlikely that your repository will assign a DOI to their version - most repositories aren't set up to issue DOIs. The repository is intended as an alternative way to access it rather than a different publication, and so material hosted by a repository tends to give the bibliographic details of the "real version" rather than provide their own.

(Also, I'm not sure what you mean by "maintaining" - are you envisaging updating it over time? This would be quite unusual for a doctoral thesis...)