Quick answers:
- You can upload to and download from BIM360 using the Revit API, afaik.
- You cannot run your Revit add-ins directly in BIM360, only in Revit itself.
- You can also make use of your Revit add-in code to create a work item and run that in the Forge Design Automation API.
https://forge.autodesk.com/api/design-automation-cover-page/
Here are some notes on how to convert a Rev it add-in to DA for Revit:
https://thebuildingcoder.typepad.com/blog/about-the-author.html#5.55
For more detailed discussions on Forge, the DA API, and how to interface with BIM360 from Forge, please refer to the questions and answers on those topics on StackOverflow:
https://forge.autodesk.com/en/support/get-help