Mendix has a module for automated unit testing. With this module it is easy to quickly run all unit tests that have been made in Mendix. This can also be done using a webservice linked to the unit test module (API).
The deployment part of mendix can indeed also be fully automated on a build server with Jenkins, but I have only tried to do this, and then gave up since automating this was not easy for me. Also the normal deployment method is so simple, automating this process was not worth it (press of a single button in the modeler).
As for test automation, I use the Unit test module of Mendix, and soapUI for testing of API's. For testing of the user interface I have made my own selenium based test tool because there is no really good off the shelf tool that I know of. You can try using selenium in combination with the unique class names that mendix conviniently adds to the different elements in you webpage.