I think you're entirely right about the state of their relationship in the past. Ignoring the Earth 2 part, they seem to be respectful colleagues about whom their is probably certain watercooler gossip in the spandex crowd, (not about a relationship, but separately, about their respective adventures---they did seem to be up on what they were each like...Dick certainly knows PG isn't the cautious type for example). But they don't seem to be all that familiar with each other.
IIRC, Kara/or Karen if you prefer, belonged to the JSA's editorial office before Flashpoint...whoever handled that. She's always sort of been "Superman Family", but in kind of the sense Jason Todd was a member of the Batman Family; the one DC itself was a bit ambivalent about. But with that ship...I don't know. You really can't move Dick entirely out of the Batman Family, and it would also be a bad idea IMO to do that to Kara.