John and Bats butted heads in the comics a few times, far more than I've ever seen Hal VS Bats anyway. Guy butts heads with everyone and of course is infamous for being laid out by Batman by picking a fight with him without his ring.
"even in Rebirth he showed great skepticism towards Hal being innocent after all."
Funnily enough that's one of the times John and Bats had a quarrel, because he was standing up for Hal and Batman was complaining about him.
Batman vs Stewart
I wish I could remember and bring up the bigger argument I've seen them have though, but it's super annoying keeping up with comic book runs and it was in a run where it was mostly focused on John going off and doing something and coming back. I think there were a few changes to his power set, but I barely remember the details apart from the fight and a spaceship.
I think the synergy thing can go with any of the GLs honestly, as you said, Kyle has synergy with Wally, John has with Vixen and depending, Hawkgirl, umm, I don't even know who has synergy with Guy....and yea they're gonna do whatever in ana adaptation anyway as you said again.
Haven't watched Young Justice. After the JLU finished, I went back to the comics and stuck to the animated movies, and those just got worse over time.