Re: limewire not installing properly?
They're not installed. Sounds like you ran Firefox from the disk image.
What you have on the desktop - the "drive icons" are disk images, a virtual disk, if you will - treat it like a CD or any other removable media. You need to open these, then, in the case of Firefox, drag the Firefox icon from there to your Application folder. With Limewire, I can't recall, the opened disk image might have an installer, or it too is a drag and drop. After that, eject the images by dragging the "drive icons" to the trash. What's left, the .dmg - you may want to keep in case you need to reinstall sometime.
Once that's done, you can start the apps from their spot in the applications folder.
"LimeWireOSX.dmg, LimeWireOSX-1.dmg" just means that you downloaded two copies.