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:
- The purpose of the trip to Germany must be plausible and comprehensible.
- The applicant must be in a position to finance his/her living and travel costs from his/her own funds or income.
- The visa holder must be prepared to leave the Schengen area before the visa expires.
- Documentary evidence must be provided of travel health insurance with a minimum coverage of 30,000 euros valid for the entire Schengen area.
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 Auswärtiges 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.