Is there good free software for creating video lessons, that allows you to make board as big as desired, highlight squares, show arrows and not checking rules, allowing to move pieces in any way, playing without king etc? I mean just board software, not software for capturing video.
You can use a setup board function is programs like scid and chessbase. These boards aren't full screen. In setup mode, they don't allow for arrows and highlights. It might be better to just program the software yourself. This would make all the features just as you like.
Perhaps http://www.apronus.com/chess/diagram/editor/ will satisfy your needs. If not, then please let me know because I am the author of this free tool and I am still working on improving it.
Perhaps http://www.apronus.com/chess/pgnviewer/ in the setup mode will be also useful. The problem with these tools is that they are very powerful but not easy to operate without instruction. So do not get discouraged just ask for help.