It's probably some of both. To do it with an MHL adapter through the charging port it's probably hardware. To run
Chromecast it's possibly software. (The app is free, and not all devices that support it are listed, so install it and see.) The Chromecast adapter itself is around for as little as $25.
As far as rooting, an update is supposed to make 2 checks before it starts updating the device - is this one of the stock ROMs I can update and has the device never been rooted? If the answer to either one is "no", the update "fails" - because updating the device could brick it. (Even if you root it, then unroot it, you could have made some changes while rooted that will cause a stock update to brick the device.)