All ATX-standard computer cases are compatible with all ATX-standard power supply units. Another issue is the selection of a PSU suitable for your computer components - what is the current consumption of your computer?
Apevia was created in 1999.
You can find various examples.Enermax,X-Dreamer II and Apevia all make them.This sort of modding is actually rather popular.
the xfit has a low profile expansion slot for like a hd5570 or hd5450 not a full sized one like the hd 5670, but if you keep the top case open you should be able to fit the card, it wont be pretty but itll work. and the power supply seems sufficient enough. you could also get a low profile version of the hd5670, the height would be fine but the width would be to wide, it needs a double low profile slot which the xfit does not have. you could modify the case/drill a hole on the side so itll fit the hd 5670 is either too wide or too tall to fit properly in the x-fit case but with some modifications you could make it work.