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Her institution and grant will bear some of the expenses of my stay, but I am not getting paid for my time there. Does this fall under a business or tourist visa category?
Since the latest harmonization under the Schengen agreement, there are no purpose-based visa catergories anymore, only short and long stay visa. The former should be what you need, and there are four prerequisites you have to meet:
Secondly, as she will remain a collaborator of mine for the foreseeable future, do I have a solid case for getting a multiple entry visa? (This would simplify my life significantly).
Yes, but I'm not sure you can get one right away. According to the AuswΓ€rtiges Amt, multiple-entry visa are typically issued to people who have proven reliable in using previous visa.
In any case, all of this is up to the German embassy or consulate which you contact to get the visa. They will tell you what they need from you.