A Volcanolosit did an analysis of the science on Trekmovie.com. Basically entering a planet's atmosphere is typically 500+ degrees hotter than a volcano so the problems suffered by the shuttle were indeed silly. However, they point out that volcanic ash can indeed be problematic so replace heat with ash and the scene works fine.
Entering a planet's atmosphere (assuming equivalent to Earth's) is only 500+ degrees hotter if you enter it from orbital speeds, around the order of 17,000 mph relative to the ground. If you enter the atmosphere at a ground speed of near zero, you don't have your hull being cooked from extreme aerodynamic heating. This is also why the ships were falling into the atmosphere - they weren't in orbit, they were pretty much stationary relative to the ground. (And also why the 3 atmospheric divers in ST09 didn't get roasted!)
As for the gym bag transporter, why couldn't it have two parts? The first part initially beams the second part to the destination, then beams the passenger. After the passenger arrives, the second part retrieves the first part from the origin point. I'm sure some fast sequencing on the part of the transporter's computer could make it seem like all one instantaneous event to an outside observer.