There are in fact several three-player chess variants. The type Sheldon designed has custom pieces (the Old Woman, the Grand Empress). These types of custom pieces are called "fairy-pieces" by Chess aficionados.
During the coffee shop scene, after Raj takes off his underwear, Sheldon can be seen smiling/smirking which is why he has his finger over his mouth at that moment.
Bernadette mentions Penny dating an Astronaut. That is a consultant, on her movie. She's filming. The Big Bang Theory uses a physicist, on set to consult with the mathematics used. Mayiam Bialik, is also used as a consultant.