This is a list of all Helvetica fonts installed to the System folder by Lion and Mountain Lion. Note that you won't find all of these as individual files. They are grouped within .dfont or .ttc fonts.
Helvetica
Helvetica Bold
Helvetica Oblique
Helvetica BoldOblique
Helvetica Light
Helvetica LightOblique
Helvetica Neue
Helvetica Neue Bold
Helvetica Neue Italic
Helvetica Neue Bold Italic
Helvetica Neue Bold Condensed
Helvetica Neue UltraLight
Helvetica Neue UltraLight Italic
Helvetica Neue Light
Helvetica Neue Light Italic
Helvetica Neue Black Condensed
Helvetica Neue Medium
There isn't a Light Condensed, or Medium Condensed, so I'm not sure why those won't load. Apple also provides no version of Gotham, so that should also work.
Do not play with the System's Fonts folder. Manually placing fonts there requires you to enter your admin password to copy them there, and again to remove them. Any fonts you do drop in that folder won't work one iota different than if you put them in the Library's Fonts folder, or the one in your user account, so there's no advantage or need to use the System folder.
If you want to manually add a font, use the one in your user account first. If it still doesn't appear in Font Book, or your apps, then the font system needs to be reset. Do the following, if necessary.
Restart your Mac and immediately hold down the Shift key when you hear the startup chime to boot into Safe Mode. Keep holding the Shift key until you see a progress bar towards the bottom of the screen. You can let go of the Shift key at that point.
OS X asks you to log in (you will get this screen on a Safe Mode boot even if your Mac is set to automatically log in). Let the Mac finish booting to the desktop and then restart normally. This will clear Font Book's database and the cache files of the user account you logged into in Safe Mode.
After that, clear all of the rest of the font cache data:
Close all running applications. From an administrator account, open the Terminal app and enter the following command. You can also copy/paste it from here into the Terminal window:
sudo atsutil databases -remove
Terminal will then ask for your admin password. As you type, it will not show anything, so be sure to enter it correctly.
This removes all font cache files. Both for the system and the current user font cache files. After running the command, close Terminal and immediately restart your Mac.