You definitely should put your tests into the repository. Tests are in my opinion part of the code and can help others immensely to understand it (if well written). Besides, they can help others when changing or contributing to your codebase. Good tests can give you the confidence that your changes do not inadvertently break anything.
The test code should be well separated from production code, though. Maven for example achieves this by putting production and test code into different folders. The question "is this file part of production or of the test code" should never arise.
I personally do not write unit tests for used libraries in my own code. I expect them to be working (at least when I use a release version, though bugs obviously can appear). It gets some test coverage in integration tests, but that's not enough.