This will be the last version of my Super Mario Bros. project for now. I've learned a huge deal and got most of the stuff to work the way I wanted it to. There are still some things missing (e.g. the invincibility star or some eye-candy like the animations for the coin blocks) so maybe I will come back to it again.
Here is the latest executable (should theoretically run under Windows, Linux and Mac as long as you have JRE 8 installed).
If someone is curious: all jars I created while I was working on this project are located here.