So I'm developing a Google Marketplace application for my company. In addition to several other features, my boss wants the app to be able to integrate with the Google Drive UI (eg be able to open files from the UI using the app). Now I understand that it's quite possible to use the Google Drive API to pull information from Drive to our Marketplace app, but is it actually possible for a Marketplace App to integrate with the Drive UI, for instance in the manner described above?
Yes, it is possible and there are many apps on market place that uses Drive UI to create or open file such as Docusign, draw.io, and gantter.
Please take a look at this GDL about integrating Drive API with Google Apps Marketplace. This will give you basic idea of how to use Dirve API with Google Apps Marketplace.
Also, take a look at documentation about Drive UI.