This DFU mode solves update problems from iOS 6 to iOS 7 or 7.1 for: iPhone 4, 4S, 5, 5C, 5S, iPad 1 2 3 4 Air and iPad Mini, Mini Retina. This solves various error messages. It does not obviously solve all errors. It is however your LAST software solution.
Error 4013: This is usually an error caused by security software on your computer. Try a friend's PC or mac and see if this helps - Just to potentially save you time, use a different USB cable as well.
Error 3194: Your hosts file is blocking connection to Apple Servers. You need to edit your hosts file (youtube how) and remove the lines referring to Apple. These usually got added when jailbreaking the iPhone and now have to be removed. Alternatively, use a different computer.