How to cite a game and props invented by the researcher?
Similarly, should there be a citation (where?) and a reference for the game (what does it look like)?


No, it is not necessary or expected.  Imagine if you replaced the game with an experiment.  Then the paper would describe the experiment/game, but not cite it.  

If the game has a published manual, you could cite that as if it were a book or report.  If there are other publications about the game, you might cite them.  This does not appear to be the case.

The fact that the game is someone's intellectual property does not imply that you need a formal citation.  You cite sources of information.  There are many examples of intellectual property which are not sources.

You might mention who invented the game.


  For example, the paper has a picture of the props, but no credit. Should there be a credit, and what does it look like?


You should credit the photographer.