When Mary does her home pregnancy test, it's the middle of the day. Back then, you had to use your first morning urine on those tests to assure accuracy.
There is a wooden, Texas-shaped plaque hanging in the school office where Sheldon's dad works. It displays the State Universities of Texas. Young Sheldon is set in the late '80s and early '90s. On the plaque, "Texas State" is visible, south of Austin, in San Marcos, TX. Prior to 2003, "Texas State University" was named "Southwest Texas State University". The name changed in 2003.