The name you give them will show as the tooltip. If you want the file to have that name, you can copy the name and paste it into the filename box after you've quit Bodyshop (notepad is a great thing
).
It's safe to rename Bodyshop files after you've exited (you probably won't be allowed to change filenames before you've quitted Bodyshop)
If you feel like doing so, you can also recathegorize and even rename clothing with the Wardrobe Wrangler if you've got it (change cathegories or ages/genders as far as it goes without needing new meshes). It's much more noob friendly than SimPE for clothing (I rarely ever us SimPE for this any longer)...
The file in the project folder is some sort of a file copy or file original (I'm not sure about this), only for Bodyshop use. It won't work ingame, not even in your own game.