There was no animosity between them whatsoever. They took pictures together. Kovalev said in one of his interviews that he is a McGregor fan. They were probably just speaking louder because of the loud music and atmosphere.
When he touched Kovalev's shoulder, he probably told him what every other boxer/fighter told him or said about: "You/Kovalev won, man".
Conor probably doesn't like Ward. Ward openly rooted for his arch nemesis Nate Diaz. Ward also seems like the complete opposite of Conor's personality. On the other hand Kovalev said that he respects Conor.
I was impressed with Conor's improved cardio in the rematch with Diaz especially his running away tactic . Conor doesn't have a big chest and likes to pack on oxygen depleting excess muscle so he has to pace himself very carefully so as to not gas himself after a round and a half . Careful steady pressure is how to take Conor out of his game at MMA easier said than done though .