How do you acctually start to play in the tournament then? Heck, I've got a new highscore while figuring that it would be used in the tournament.

Pretty much the same here.

You go to the tournament link then follow some other links to start playing

I can't get it working. All that happens is that I play the game and then.. nothing, it won't even save me score and it doesn't matter if it's good or better.. HOHO!