You can get a new one but you can also sync your controllers by using the USB port in the back of the Xbox One mine is broken as well so I just use the USB port to sync my controllers to the Xbox
Sounds like a hard drive issue my friend has the same issue his stays on the green screen but does not give him the option to factory reset so I have to go in and turn in early and do a factory reset and it happens to him every other week it's weird because it randomly happens but everything saves well and all that so I'm still trying to troubleshoot only thing I have for right now is every time it happens just do a factory reset and keep your games and apps
Your HDMI port is bad and has to be replaced try holding on to the power button eject button and sync button at the same time until you hear two beeps an option to do a factory reset should come up on the screen if nothing happens but you still hear two beeps you might have a hard drive issue or an HDMI port is damaged