Is there any way I can make anything that is currently shown on the screen of an iPhone 5 to be displayed on a regular TV or projector that has the usual HDMI, composite video, DVI, DSub inputs via a simple cable or adapter and without Airplay/Apple TV?
At the moment, this is currently not possible as existing adapters are not compatible with the Lightning connector (even with current converters). See this link for more information.
At the Apple event on the 23rd October, two new adapters were announced VGA to Lightning and Lightning Digital AV (HDMI). If you read the details on the store page carefully, they both say they support mirroring (in the overview section).
Apple is now showing Lighting to hdmi and vga and can be found on the link below.