When Richard Thomas wins the Best Actor award at the 1973 Emmys, you can see Sian Barbara Allen sitting next to him per YouTube video. They were in a relationship at the time.
At 16:55, the program the family is listening to is likely "The Fire Chief," a popular radio show that aired in the early 1930s. The show was hosted by comedian Ed Wynn, whose distinctive voice can be heard quite clearly in this scene.
The ballad John-Boy sings to Jenny while playing the dulcimer is "The Turtle Dove," by John Jacob Niles (1892-1980), an American composer, singer and collector of traditional ballads. Niles was an important influence on the American folk music revival of the 1950s and 1960s.
Music of the Appalachian dulcimer is featured in this episode. There is a a popular traditional Appalachian tune called Barbara Allen, though it is not played in this episode. Ironically, Sian Barbara Allen is the name of the actress playing the character for whom John-Boy plays his dulcimer.