Testing yes, but how?

When people start looking into the testing domain, very similar questions arise:
What to test? And more important, what not? What should I mock? What should I test with unit tests and what with Instrumentation? Do I need Robolectric? When? Do we need an in-memory database?
Let’s look at some examples.

SoundCloud has a huge base of tests, unit, but also UI tests. For a 13y old codebase that is very important and a huge safety net for keeping the team working and onboard new developers. But we also learned what is working well and what should be done better.
The author is a huge fan of TDD and gave various testing and TDD sessions in the past. But it looks like some questions are still unanswered.