Nope. It has more stability than the Mustang, and is more nimble than the Challanger.
All 3 are planning on smaller sizes.
Depends on a lot of variables here.
Tweaks to the Mustang's chassis since it's initial release in '05 have made it a substantially better handling car than it once was. Although people love to complain about the live rear axle, as I recall, the most-recent updates for '13 made it almost unremarkable.
The Challenger, circa 2009, had a pretty reasonable setup, but the Mercedes designs made it quite large and quite heavy. Diamler-Chrysler made it a point not to sell the car as a sporty option, it seemed to be meant only as a cruiser with some serious muscle under the hood. As much as SRT would love to tout the fact that it's a capable car, it really isn't.
The Camaro on the other hand has the most "modern" chassis of the the three, but it came out of the very end of the "old GM" lifespan. It's heavy, very heavy, but it's a true independent rear suspension, multiple links all over, and all that jazz. Early Camaros were hammered for being a bit vague feel wise, although I believe it's been updated quite a bit, especially for the '14 model.
Like most cars these days, your trim level will determine performance. On the whole, yes, I'd agree with you. But, models like the 1LE, GT Track Pack, and so on, turn things around a bit.