The episode of Star Trek (1966) Sheldon is watching is from the original series, Plato's Stepchildren (1968). The episode is noted for featuring the first interracial kiss on television between Capt Kirk (William Shatner ) and Lt Uhura (Nichelle Nichols ). The episode first aired November 22, 1968.
Sheldon tells Paige that his birthday is on February 26th.
Though Paige is introduced as a month younger than Sheldon, Mckenna Grace is actually two years older than Iain Armitage.
The phone number that John recites that belongs to the Coopers (409-356-6049) indicates that they live in Galveston, TX.