Miguel mentions Peter Cetera's The Glory of Love, which was featured in the film Karate Kid II.
Bicephaly (Greek for "condition of having two heads") is a reference to a genetic condition that happens in utero, where, because the embryo doesn't split correctly or completely, it evolves into a fetus with two heads being supported by a single torso. This phenomenon is incredibly rare, but happens in both humans and animals to differing extents. In this case it's a coded reference to the two-sensei model at each of the dojos, and the clashes between them.
Nate and Bert start hassling Kenny directly in front of a poster that reads "STOP BULLYING".
Miguel tricking Cobra Kai into being drenched by sprinklers recalls Daniel soaking Johnny in the first film. Johnny's reaction implies that he makes this connection and is troubled by it, while Daniel praises Miguel for finding an alternative to fighting.
Silver repeats his famous line of "A man can't stand, he can't fight", this time in a more benign context.