No, Jenkins is just used as one example of an automation server. Any automation server of your choice should work fine, as long as it can execute the LaunchTcUnit.bat windows batch script or the TcUnit-Runner.exe windows executable, and it can handle standard  xUnit/JUnit XML format it should work just fine.

Required TcUnit version: 1.1 or later

Yes, this is possible with the limitation that it is not possible to run TwinCAT on a Windows/shared core but instead it is required that TwinCAT (and more specifically the unit-tests) are to be run on an isolated core in the virtual machine. Go here to read more on the subject.

Required TcUnit version: 1.0 or later

Check the console output in Jenkins if it provides any information. Also, try to build the project manually (by opening Visual Studio) on the build server and check if builds correctly.