No. The Lightning to 30-pin adapter does not support video out. It does support analog audio output, USB audio, as well as syncing and charging.
Yes, the Lightning to 30-pin adapter supports analog audio output, USB audio, as well as syncing and charging. It does not support video output.
The 30-pin adapter is available for $29 from the Apple Store. The iPad 5, 5s 5c, 4th generation iPad Air, iPad mini, iPad mini with Retina display, 7th generation iPad nano and 5th generation iPod touch all use the Lightning connector.