The Fit has a very low cargo floor with the seat folded, so the Dog might be right behind the driver, but lower. It is no replacement for an SUV as a dog carrier, but it would be the best small hatch for the job. That said, the CX-5 would be excellent. It’s not fast, but car magazines praise its sharp handling. The Kia Soul would definitely make my list. It has a big, boxy cargo area that would be good for a big dog. It also treats human passengers well, front and rear. Another frugal model is the Scion xB, similarly comfy for both people and stuff. It doesn’t sell well, so if you like it push hard on the price. The just discontinued Toyota Matrix is a very good choice. They even made the cargo area of ribbed plastic, easy to clean. Mechanically it’s close to a Corolla and very solid. There are any number of other small crossovers, but the Mazda CX-5, the CR-V and the Toyota RAV4 are three of the best and the Mazda seems to most effectively combine sporty handling, great fuel economy, and nice styling. It’s also likely to be fairly reliable, like other recent Mazdas.
One other worth a look is the just out 2014 Nissan Rogue. Previous Rogues have been homely and cheap-feeling, but this new one is completely different, pleasant to look at inside and out and reputedly decent to drive. Reliability is hard to guess. Nissan is often a step behind the better Asian makes, but they have made some very good cars, too. Worth a look if none of the others please you.