Responsive handling and ride quality are generally mutually exclusive, but it's not a concrete rule. I rented a 2014 Mazda3 5-Door 2.5L a few weeks ago, I found the ride to be very good for its handling capability. I think the new ones are a bit pricey but they're a pretty good car.The Mazda6 is just a bigger and more expensive Mazda3.
Although I'm a subaru nut and can tell you exactly how to phenominally increase handling and shift response for only a few hundred dollars in modifications and labor, or how to smoothen a WRX ride a bit, they're overall meh commuter cars unless you need the AWD, in which case they're some of the best.
Ever driven a Ford? The recent fords are pretty stunning in terms of handling and economy. I believe that their dual-clutch auto-manual comes standard, and it's pretty impressive. Not DSG-level but close. The chassis are extremely well-sorted, and the ride is not bad. I doubt it's much smoother than the Mazdas.
I'd also recommend trying a Golf, or a Jetta if the Golf's too small. You'll find great quality, handling, and transmissions there. It'll be a bit pricer, and not sure about the rides, as I haven't driven one recently.