Cleanup tests: "merge" `test/warnings` into `test/doctests` (possibly renaming the latter), and more

[fingolfin]

As discussed with @mortenpi on Slack. Excerpts:

But it certainly would make sense to move my new warning tests under the test/doctests "test framework" -- but perhaps renamed (I thought test/doctests was purely for testing doctests so didn't even look there). Perhaps test/output_tests and then have subdirs like doctests, warnings ? and perhaps a small test/README.md that gives some hints on where to add certain kinds of new tests?

it might be good if we can identify a few other "groups" of "similar" tests, and move them into other test/... subdirs. (anything we can not categorize / which seems like it should be redone, could go into a test/legacy_tests dir -- just to make clear to newcomers "don't imitate those tests"

We also have those example manual tests that are messy.. but they are more integration-test stuff.