Is it possible to downgrade Windows 8 Enterprise to a Windows 8 Pro? If so, do I keep the settings (such as the Proxy Settings on the Control Panel) or do I need to reinstall my (non-metro) apps?
No, you can not downgrade from enterprise to pro without re-installing windows. (also it would be a upgrade, as enterprise and pro are identical, except pro also has media center)
If you are trying to go from Enterprise Eval to something else here is some more information about it.