Don't know the Droid Turbo specifically, but many many android phones and tablets will play audio out through their usb port via an "On-the-Go" usb cable. Not any old usb cable that physically plugs into your device, cable must be labelled as an "on-the-go". Your device must also be able to use on-the-go. You will then need some sort of DAC to connect to the phone, plus an app on your device that will utilise your device's ability to send music out through the on-the-go cable.
Check out Enable USB audio on any Android 4.0 smartphone/tablet - AVS Forum | Home Theater Discussions And Reviews
As for video you need to buy what is called an MHL cable that has a plug on one end that fits your phone's usb port, and it has an HDMI socket on the other end which you can then hook up to your tv or, if you have one, you av receiver. Google MHL.