Good post. I was in the same situation about 3 years ago and the transition from waterfall to agile was tricky. I encountered many pain points in the move but once I overcame them and my role had changed I realised that this way of working really suits testing.
The common myth that testers are not required is easily dispelled.
1. While a developer is coding a task, it is impossible for a tester to test it (it doesn't exist yet). What then is the role of a tester at this point
In my experience the tester could be working with the customer to fine tune the stories in the sprint.
They are usually working with the developers to fine tune the code that they are delivering. i.e. advising on edge cases, flows, errors etc.
They can often be involved in designing the tests that the coder will write to perform TDD.
If the agile team is fairly advanced then the tester would normally be writing the ATDD (Acceptance Test Driven Development) tests. These could be in a tool such as Fitnesse or Robot Framework or they could be more advanced ruby tests or even some other programming language. Or in some cases, simple record and playback can often be beneficial for a small number of tests.
They would obviously be writing tests and planning some exploratory testing scenarios or ideas.
The tricky thing to comprehend sometimes for the team is that the story does not have to be complete in order to drop it to the test stack for testing. For example the coders could drop a screen with half of the fields planned on it. The tester could test this half whilst the other half is being coded and hence feedback in with early test results. Testing doesn't have to take place on "finished" stories.
2. Is the tester now involved in unit testing? Is this done parallel to black box testing?
Ideally the coders would be doing TDD. Writing the test and then writing the code to make the test pass. And if the coders are wanting really good TDD then they would be liasing with the tester to think up the tests.
If TDD is not being done then the coders should be writing unit tests at the same time as coding. It probably shouldn't be an after thought or after task after the software has been dropped. The whole point of tests is to test the software is correct to avoid wasting time later down the line. It's all about instant feedback.
3. What does the tester do during a sprint where primarily infrastructural changes have been made, that may only be testable in unit testing?
Ideally the tester would be working with the team and the customer (who by the way, is part of the team!) to define the planned stories and build in some good, detailed acceptance critiera. This is invaluable and can save loads of time later down the line. The tester could also be learning new automation techniques, planning test environments, helping to document the outcome of the planning.
Ideally each story in the sprint would be testable in some way, shape or form. This doesn't mean it should be by the test team, but should be testable. So the tester could be working with the rest of the team working out how to make sure stories are testable.
I post some agile tips here : http://thesocialtester.posterous.com/
Hope this helps you out
Rob..