The musical number which The Monkees sing a cappella is "Riu Chiu", a Christmas carol written in archaic Spanish. This carol's printed history has been traced to 1556, and may even be much older than that.
When Micky Dolenz, dressed as Santa (or: Father Christmas as Englishman Davy Jones calls him) and Davy as his elf helper appear on the roof, Davy is wearing one of Michael Nesmith's green wool hats. Not his original darker green or his: 'newer-hipper-double-breasted-six-button' version, but the 'classic' green. Mike had given up wearing his wool hat for a few episodes, but it is back on top for the last time in The Devil and Peter Tork (1968).
As a sign of the word "gay" meaning "homosexual" (and the fact its popular usage as such was coming on in those days), Davy and Mickey both "strike a pose" when they sing the word.
Butch Patrick, who plays Melvin, is better known to audiences as Eddie Munster from The Munsters, a show, which had finished its run in 1966, the year The Monkees started. He is here about 14 years old.