At my company we use and endorse Agile. Our QA team members are involved in unit test creation, maintaining the regression testing infrastructure and, just like in waterfall, they also test each feature upon completion.
When doing infrastructural changes, they also participate to make sure that the new infrastructure is testable.
So, from my limited experience, I'll try to answer your points:
- If there's nothing to test yet, start setting up a regression/testing infrastructure and make sure that whatever is being done will be testable
- Yes, he may do both
- Maintains the testing infrastructure and hunts whoever breaks the tests
