Old news.
Because Callista never caught him balls deep inside Yvette.
Yes when Veronica was approximately 10 years old, Kane being elementary school age (confirmed by Tess btw, don't bother arguing this).
Why they met could have been for numerous reasons (and the Kane/Veronica meeting could have been completely by chance and unintended as a by-product of just being dragged along), the actual meeting was Callista/Yvette, so at least at that point in time, there was no bad blood, or if there was, not enough to not be able to communicate and deal with something. However, this was years later after the fact, Maybe Callista put it behind her for the sake of discussing whatever it was that needed to be discussed.
You're confusing two events as one:
Park 1: Ep. 3 flashback. Callista has cut Kane's playground time short because they got to get home in time for Olivia's recital. In such a hurry, Callista almost leaves him behind, then apologizes for getting too far ahead of him. Kane then asks why Callista was talking to a woman and girl "with fiery hair", whose name he apparently does not know.
Park 2: Ep 6 Park flashback. 10yo Veronica is bullied by other kids in the park and hides by a large statue alone. The flashback begins with Kane upset with, and actively avoiding, his mother because of her strictness and making him do so much extra study work (2 assignments from school + 5 from Callista). In his anger-infused wondering he happens upon the distraught Veronica. While he himself is not interested in the play area at the moment, he asks the the lonely girl while she isn't playing there herself, which she says that was because of the other kids bullied her. Having not been at the play area, and therefore not seen the event, Kane incorrectly guesses the single bully's identity, and Veronica corrects him that nearly all kids were bullying her. In a rare display of the Cole-Callista intelligence, Kane deduces that this mystery girl is a Cape Town native from her wristwatch. Veronica then gives her name as "Stacy", and Yvette pages her.
Kane's attitude toward Callista in each flashback is very different, and his speech patterns also different. He says "playground" in Ep. 3 but "play area" in Ep. 6, signifing either a different place or a different time (i.e. he's changed his vocabulary). In the first flashback, he doesn't know "Stacy's" name, and in the second flashback, he was not spending any time in the playground/play area, and it doesn't seem like he sees Yvette.
As for the first flashback where the meeting took place, there are 3 possible scenarios on how this could have happened. 1) Yvette and Callista were close, and they met up to catch up, and Yvette wanted to know why Callista left. 2) they didn't choose to meet, Yvette was just a messenger from Alexander, and he certainly couldn't have risked letting Elena possibly see Olivia, so Yvette was the only option. 3) The meeting was purely by chance.
20 years is a long time to develop a blood feud, so maybe Yvette's hatred for Callista started at the Breakup, or the Leaving, or after years of Alexander blaming her for not measuring up to Callista, OR perhaps it's after the second flashback when Veronica, at some point, reached out to Callista, thus she became more Veronica's mother than Yvette could ever hope to be, due to Callista's mind being the only one Veronica could respect and admire (Ep. 5 ninja meeting). Either way, there's evidence of it in the present, and just like the death of Good Alexander, or D/J/A/A's births, we don't have enough information to pinpoint exactly where in the timeline these events took place.