During the band's song at the fair, in a rare change of pace, in addition to seeing the group perform the song, you also see the Partridge Family off stage "music video style" singing in various scenarios, like David singing to a line of girls, etc. This is very different than the usual strict performance clip on stage or in their garage rehearsing.
The group of horses seen in one of the scenes are Suzanne Crough's (Tracy) actual horses.
This show leads the viewer to believe that horses only sleep at night and that they sleep through the night. However, horses tend to sleep in short increments day and night. Horses will lie down if they are sick or very worn out, but they typically sleep standing up. This is possible due to something in their legs called a stay apparatus that locks their legs, allowing then to remain standing. A horse might sleep 15-30 minutes at a time during the day and night.
The song Danny plays in an attempt to get the horse to fall asleep is "Moonlight Serenade" performed by the Glen Miller Orchestra.
The song the family performs at the fair is "Breaking Up Is Hard to Do," co-written by Neil Sedaka. Sedaka's two recordings in 1962 and 1975 are probably the most famous versions of the song but it has been covered by many established performing acts.