I would indeed recommend a better power supply than 475 W. The R9 280X will eat up about 250 watts on its own, and you're likely going to end up running it at its full capacity. Additionally, most R9 280X's will likely require at least one 6-pin and one 8-pin PCIe power connector, if not two 8-pins. Some lower-end PSUs may not have those available, as they aren't intended to be used with higher-end video cards.
There's a lot of options, but I think you'd be in good shape with this.
As for the 280X's length, that depends on the specific product you get. An R9 280X from Asus/MSI/Sapphire etc. all may be different sizes. I'd recommend checking the manufacturer specifications for your case, as most list a maximum supported video card length. Newegg usually lists the length of each card.