Diana is (a tad) overpowered in my opinion. I find the things causing that being her heavily damaging, low cd Q and her passive, the 3-hit proc, that can just add a whole lot more to her spike(s). Get her a little bit of AP and she can already do heavy spikes while still building tanky. Guess it's the problem that Ahri once had too, being too damaging while still building tanky. As for evading her Q? That's actually the easiest part of playing vs Diana. Move to Diana's left or away from her when she casts it and you'll easily dodge it.
As for her role, it obviously is, as said before, spiking the main carry and then letting the team pick it up. In essence you can compare it to what any assassin is supposed to do, spiking a high-priority target down to turn a fight in your favor (though Diana is way better at it as is due to her Q, W, R, Ignite/AA, R combo combined with VERY strong AP scaling).
Countering her is the hardest thing at the moment, though in my opinion she is the same as Darius in that regard. Deny her farm and kills if she lanes, when she jungles, counter-jungle and do NOT let her get any kills. She snowballs like mad. In a fight, once she goes in, she ain't going out until either her (team) or her opponent(s) are dead. Any hard CC that interrupts or stops abilities will shut her down, bigtime. Knock-ups, Silence, even Taunts will mess her up heavily. (Alistar or Cho Gath, for instance, both do quite well.)