You summed up the kind-of-flirting thing SO WELL there, just want to point that out. I think with Coop he does this kind of semi-flirting thing with people to both put them at ease (because you can get the job done a lot quicker if people like you) and to keep them on their toes just a little bit, ESPECIALLY in Johto where he gets to indulge in his playful side a lot more than he does in canon. It's a quick way of striking up a connection to work on - but yeah, it's easy to get serious. And the fun part with Carmen for Coop is exactly that, that she throws it back at him so easily but understands immediately when to drop it and discuss serious matters.
They're just really similar like that and it's a lot of fun.

ALSO re: Coop and Yagyuu I had no idea that him having Phantom's daughter was a special thing! And it makes me aaw a little. Because yeah, they're not exactly friends, but I like that there seems to be a mutual respect behind the intelligent discussions they've got so far and I'm enjoying it. c: