Diana is not that strong in my opinion.
She is melee, but not very tanky, so she should die quickly when focused. She is a lot easier to gank than most AP mids.
In the jungle she is pretty strong, but she will fall off lategame, because she can't put out as much damage and she isn't that tough.

Also note even though a champion can be countered, that does not mean he will be countered. People prefer to ban a champion that needs to be countered and play something they like.

I just played against her and even with our Jax counter jungling her and taking her buffs, she still became a menace late game. The amount of health she can get and still have quite a high burst damage combo on our carry + team cc meant it was either focus Diana and die to their team or focus their carry and die to Diana. It was stupid.

Diana is op as **** imo.
Her shield makes her incredibly strong.

Her sweep is too op, hardly to evade. That sweep bursts away 200 damage in one go on Ahri. Her ulti lets her dash to me without problem and her third spell knocks me just to an area.
Her shield is also op.

GJ RIOT. Nice balanced champ of course!

She is a burst caster so while she is easy to beat early game she will still become a terror at the 25 minute mark all she needs is 2 tanky people on the team (support and toplane/jungler) and she will clean your whole team in a matter of seconds.
I dont think this is over powerd as this is how she it ment to be played but I will still ask for her to be banned every game.

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.)