I think a 'cross bike will do anything you want it to but if you start chasing other people you might want to get a road bike. I did have a friend who did all the big climbing rides on a cx bike but finally got a road bike.
CX bike could come with stronger wheels and that should be a concern. I think any of those bikes would work but some of them might come with light/cheap wheels.
I've broken enough wheels and you will too if you ride a lot.

Oh, the bars going down to the rear wheel are called "seatstays".