Does the Moto X have DLNA? If not, is there an app that gives it the capability of communicating with a DLNA music system?
No, unfortunately, Moto X doesn't have DLNA. I just got mine yesterday and was trying to figure that out myself. That was a feature I didn't verify, assuming it would have it. It does work with Miracast, which mirrors your phone's screen on the TV screen. And, it does work with Chromecast for the few apps that support that now. (with tab cast feature from a desktop, you have a little extra ability, with sharing a browser window, and even (experimentally) your desktop, but only from a PC running Chrome browser, not mobile Chrome). But, as other posters mentioned, you will have to find a DLNA app on the Play Store for true "DLNA" uses, where you can wirelessly access your media like a harddrive and play them without relying on the phone except to serve the media files. There are a lot. Just check reviews. The device you have as your DLNA receiver (renderer) is important, too. I have an Xbox I use with my Droid Bionic, and it works still with DLNA apps with my Moto X. With both phones, DLNA is cludgy...often I have to reboot my phone and Xbox for the video to play smoothly, but then it does as long as I don't try to rewind. Good luck! I did read that (at least) 2 major Android devs are working on ways to use Chromecast to play local videos and videos from Google Drive. But, possibly won't ever be available for stock Android because of trying to protect DRM material from being mis-shared.